In physics, spacetime is any mathematical model that fuses the three
dimensions of space and the one dimension of time into a single
four-dimensional continuum.
Contents 1 Introduction 1.1 Definitions 1.2 History 2
2.1
2.7.1 Mutual time dilation 2.7.2 Twin paradox 2.8 Gravitation 3 Basic mathematics of spacetime 3.1 Galilean transformations
3.2 Relativistic composition of velocities
3.3
3.4.1 Deriving the Lorentz transformations 3.4.2 Linearity of the Lorentz transformations 3.5 Doppler effect 3.5.1 Longitudinal Doppler effect 3.5.2 Transverse Doppler effect 3.6 Energy and momentum 3.6.1 Extending momentum to four dimensions
3.6.2
3.7 Conservation laws 3.7.1 Total momentum 3.7.2 Choice of reference frames 3.7.3 Energy and momentum conservation 4 Beyond the basics 4.1 Rapidity 4.2 4‑vectors 4.2.1 Definition of 4-vectors 4.2.2 Properties of 4-vectors 4.2.3 Examples of 4-vectors 4.2.4 4-vectors and physical law 4.3 Acceleration 4.3.1 Dewan–Beran–Bell spaceship paradox 4.3.2 Accelerated observer with horizon 5 Introduction to curved spacetime 5.1 Basic propositions 5.2 Curvature of time 5.3 Curvature of space 5.4 Sources of spacetime curvature 5.4.1 Energy-momentum
5.4.2
5.4.3.1 • Active, passive, and inertial mass
5.4.3.2 •
6 Technical topics 6.1 Riemannian geometry 6.2 Curved manifolds 6.3 Privileged character of 3+1 spacetime 7 Section summaries 7.1 Introduction summary
7.2
8 See also 9 Notes 10 Additional details 11 References 12 Further reading 13 External links Introduction[edit] Definitions[edit]
Click here for a brief section summary
Non-relativistic classical mechanics treats time as a universal
quantity of measurement which is uniform throughout space and which is
separate from space.
c displaystyle c (conventionally called the speed-of-light) relates distances measured
in space with distances measured in time. The magnitude of this scale
factor (nearly 7008300000000000000♠300,000 km in space being
equivalent to 1 second in time), along with the fact that
spacetime is a manifold, implies that at ordinary, non-relativistic
speeds and at ordinary, human-scale distances, there is little that
humans might observe which is noticeably different from what they
might observe if the world were Euclidean. It was only with the advent
of sensitive scientific measurements in the mid-1800s, such as the
Figure 1-1. Each location in spacetime is marked by four numbers defined by a frame of reference: the position in space, and the time (which can be visualized as the reading of a clock located at each position in space). The 'observer' synchronizes the clocks according to their own reference frame. In special relativity, an observer will, in most cases, mean a frame
of reference from which a set of objects or events are being measured.
This usage differs significantly from the ordinary English meaning of
the term. Reference frames are inherently nonlocal constructs, and
according to this usage of the term, it does not make sense to speak
of an observer as having a location. In Fig. 1‑1, imagine that
the frame under consideration is equipped with a dense lattice of
clocks, synchronized within this reference frame, that extends
indefinitely throughout the three dimensions of space. Any specific
location within the lattice is not important. The latticework of
clocks is used to determine the time and position of events taking
place within the whole frame. The term observer refers to the entire
ensemble of clocks associated with one inertial frame of
reference.[6]:17–22 In this idealized case, every point in space has
a clock associated with it, and thus the clocks register each event
instantly, with no time delay between an event and its recording. A
real observer, however, will see a delay between the emission of a
signal and its detection due to the speed of light. To synchronize the
clocks, in the data reduction following an experiment, the time when a
signal is received will be corrected to reflect its actual time were
it to have been recorded by an idealized lattice of clocks.
In many books on special relativity, especially older ones, the word
"observer" is used in the more ordinary sense of the word. It is
usually clear from context which meaning has been adopted.
Physicists distinguish between what one measures or observes (after
one has factored out signal propagation delays), versus what one
visually sees without such corrections. Failure to understand the
difference between what one measures/observes versus what one sees is
the source of much error among beginning students of relativity.[7]
Return to Introduction
History[edit]
Main articles:
Figure 1-2. Michelson and Morley expected that motion through the aether would cause a differential phase shift between light traversing the two arms of their apparatus. The most logical explanation of their negative result, aether dragging, was in conflict with the observation of stellar aberration. By the mid-1800s, various experiments such as the observation of the
Hendrik Lorentz Henri Poincaré Albert Einstein Hermann Minkowski Figure 1-3. An important example is Henri Poincaré,[12][13]:73–80,93–95 who
in 1898 argued that the simultaneity of two events is a matter of
convention.[14][note 2] In 1900, he recognized that Lorentz's "local
time" is actually what is indicated by moving clocks by applying an
explicitly operational definition of clock synchronization assuming
constant light speed.[note 3] In 1900 and 1904, he suggested the
inherent undetectability of the aether by emphasizing the validity of
what he called the principle of relativity, and in 1905/1906[15] he
mathematically perfected Lorentz's theory of electrons in order to
bring it into accordance with the postulate of relativity. While
discussing various hypotheses on Lorentz invariant gravitation, he
introduced the innovative concept of a 4-dimensional space-time by
defining various four vectors, namely four-position, four-velocity,
and four-force.[16][17] He did not pursue the 4-dimensional formalism
in subsequent papers, however, stating that this line of research
seemed to "entail great pain for limited profit", ultimately
concluding "that three-dimensional language seems the best suited to
the description of our world".[17] Furthermore, even as late as 1909,
Poincaré continued to believe in the dynamical interpretation of the
Lorentz transform.[10]:163–174 For these and other reasons, most
historians of science argue that Poincaré did not invent what is now
called special relativity.[13][10]
In 1905, Einstein introduced special relativity (even though without
using the techniques of the spacetime formalism) in its modern
understanding as a theory of space and time.[13][10] While his results
are mathematically equivalent to those of Lorentz and Poincaré, it
was Einstein who showed that the Lorentz transformations are not the
result of interactions between matter and aether, but rather concern
the nature of space and time itself. Einstein performed his analyses
in terms of kinematics (the study of moving bodies without reference
to forces) rather than dynamics. He obtained all of his results by
recognizing that the entire theory can be built upon two postulates:
The principle of relativity and the principle of the constancy of
light speed. In addition, Einstein in 1905 superseded previous
attempts of an electromagnetic mass-energy relation by introducing the
general equivalence of mass and energy, which was instrumental for his
subsequent formulation of the equivalence principle in 1907, which
declares the equivalence of inertial and gravitational mass. By using
the mass-energy equivalence, Einstein showed, in addition, that the
gravitational mass of a body is proportional to its energy content,
which was one of early results in developing general relativity. While
it would appear that he did not at first think geometrically about
spacetime,[18]:219 in the further development of general relativity
Einstein fully incorporated the spacetime formalism.
When Einstein published in 1905, another of his competitors, his
former mathematics professor Hermann Minkowski, had also arrived at
most of the basic elements of special relativity.
“
I went to Cologne, met Minkowski and heard his celebrated lecture
'
Minkowski had been concerned with the state of electrodynamics after
Michelson's disruptive experiments at least since the summer of 1905,
when Minkowski and
Figure 1-4. Hand-colored transparency presented by Minkowski in his 1908 Raum und Zeit lecture. On November 5, 1907 (a little more than a year before his death),
Minkowski introduced his geometric interpretation of spacetime in a
lecture to the Göttingen Mathematical society with the title, The
Relativity Principle (Das Relativitätsprinzip).[note 4] On September
21, 1908, Minkowski presented his famous talk,
Δ d displaystyle Delta d between two points can be defined using the Pythagorean theorem: ( Δ d ) 2 = ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 displaystyle left(Delta d right) ^ 2 = left(Delta x right) ^ 2 + left(Delta y right) ^ 2 + left(Delta z right) ^ 2 Although two viewers may measure the x,y, and z position of the two
points using different coordinate systems, the distance between the
points will be the same for both (assuming that they are measuring
using the same units). The distance is "invariant".
In special relativity, however, the distance between two points is no
longer the same if measured by two different observers when one of the
observers is moving, because of the Lorentz contraction. The situation
is even more complicated if the two points are separated in time as
well as in space. For example, if one observer sees two events occur
at the same place, but at different times, a person moving with
respect to the first observer will see the two events occurring at
different places, because (from their point of view) they are
stationary, and the position of the event is receding or approaching.
Thus, a different measure must be used to measure the effective
"distance" between two events.
In four-dimensional spacetime, the analog to distance is the interval.
Although time comes in as a fourth dimension, it is treated
differently than the spatial dimensions.
Δ t displaystyle Delta t and a spatial distance Δ x displaystyle Delta x . Then the spacetime interval ( Δ s ) 2 displaystyle left(Delta s right) ^ 2 between the two events that are separated by a distance Δ x displaystyle Delta x in space and by Δ c t = c Δ t displaystyle Delta ct =cDelta t in the c t displaystyle ct -coordinate is: ( Δ s ) 2 = ( Δ c t ) 2 − ( Δ x ) 2 displaystyle (Delta s)^ 2 =(Delta ct)^ 2 -(Delta x)^ 2 , or for three space dimensions, ( Δ s ) 2 = ( Δ c t ) 2 − ( Δ x ) 2 − ( Δ y ) 2 − ( Δ z ) 2 displaystyle (Delta s)^ 2 =(Delta ct)^ 2 -(Delta x)^ 2 -(Delta y)^ 2 -(Delta z)^ 2 [25] The constant c displaystyle textrm c , the speed of light, converts the units used to measure time (seconds) into units used to measure distance (meters). Note on nomenclature: Although for brevity, one frequently sees interval expressions expressed without deltas, including in most of the following discussion, it should be understood that in general, x displaystyle x means Δ x displaystyle Delta x , etc. We are always concerned with differences of spatial or temporal coordinate values belonging to two events, and since there is no preferred origin, single coordinate values have no essential meaning. Figure 2-1.
The equation above is similar to the Pythagorean theorem, except with a minus sign between the ( c t ) 2 displaystyle ( textrm c ,t)^ 2 and the x 2 displaystyle x^ 2 terms. Note also that the spacetime interval is the quantity s 2 displaystyle s^ 2 , not s displaystyle s itself. The reason is that unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative. Rather than deal with square roots of negative numbers, physicists customarily regard s 2 displaystyle s^ 2 as a distinct symbol in itself, rather than the square of something.[18]:217 Because of the minus sign, the spacetime interval between two distinct events can be zero. If s 2 displaystyle s^ 2 is positive, the spacetime interval is timelike, meaning that two events are separated by more time than space. If s 2 displaystyle s^ 2 is negative, the spacetime interval is spacelike, meaning that two
events are separated by more space than time.
x = ± c t displaystyle x=pm textrm c ,t . In other words, the spacetime interval between two events on the world line of something moving at the speed of light is zero. Such an interval is termed lightlike or null. A photon arriving in our eye from a distant star will not have aged, despite having (from our perspective) spent years in its passage. A spacetime diagram is typically drawn with only a single space and a single time coordinate. Fig. 2‑1 presents a spacetime diagram illustrating the world lines (i.e. paths in spacetime) of two photons, A and B, originating from the same event and going in opposite directions. In addition, C illustrates the world line of a slower-than-light-speed object. The vertical time coordinate is scaled by c displaystyle textrm c so that it has the same units (meters) as the horizontal space coordinate. Since photons travel at the speed of light, their world lines have a slope of ±1. In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time. Note on nomenclature: There are two sign conventions in use in the relativity literature: s 2 = ( c t ) 2 − x 2 − y 2 − z 2 displaystyle s^ 2 =( textrm c t)^ 2 -x^ 2 -y^ 2 -z^ 2 and s 2 = − ( c t ) 2 + x 2 + y 2 + z 2 displaystyle s^ 2 =-( textrm c t)^ 2 +x^ 2 +y^ 2 +z^ 2 These sign conventions are associated with the metric signatures (+ − − −) and (− + + +). A minor variation is to place the time coordinate last rather than first. Both conventions are widely used within the field of study. Return to Introduction Reference frames[edit] Click here for a brief section summary Figure 2-2. Galilean diagram of two frames of reference in standard configuration. Figure 2-3. (a) Galilean diagram of two frames of reference in standard configuration. (b) spacetime diagram of two frames of reference. (c) spacetime diagram showing the path of a reflected light pulse. In comparing measurements made by relatively moving observers in different reference frames, it is useful to work with the frames in a standard configuration. In Fig. 2‑2, two Galilean reference frames (i.e. conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame S′ (pronounced "S prime") belongs to a second observer O′. The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame S′. Frame S′ moves in the x-direction of frame S with a constant velocity v as measured in frame S. The origins of frames S and S′ are coincident when time t = 0 for frame S and t′ = 0 for frame S′.[3]:107 Fig. 2‑3a redraws Fig. 2‑2 in a different orientation. Fig. 2‑3b illustrates a spacetime diagram from the viewpoint of observer O. Since S and S′ are in standard configuration, their origins coincide at times t = 0 in frame S and t′ = 0 in frame S'. The ct′ axis passes through the events in frame S′ which have x′ = 0. But the points with x′ = 0 are moving in the x-direction of frame S with velocity v, so that they are not coincident with the ct axis at any time other than zero. Therefore, the ct′ axis is tilted with respect to the ct axis by an angle θ given by tan θ = v / c . displaystyle tan theta =v/c. The x′ axis is also tilted with respect to the x axis. To determine
the angle of this tilt, we recall that the slope of the world line of
a light pulse is always ±1. Fig. 2‑3c presents a
spacetime diagram from the viewpoint of observer O′. Event P
represents the emission of a light pulse at x′ = 0,
ct′ = −a. The pulse is reflected from a mirror situated
a distance a from the light source (event Q), and returns to the light
source at x′ = 0, ct′ = a (event R).
The same events P, Q, R are plotted in Fig. 2‑3b in the frame
of observer O. The light paths have slopes = 1 and −1
so that △PQR forms a right triangle. Since
OP = OQ = OR, the angle between x′ and x must
also be θ.[3]:113–118
While the rest frame has space and time axes that meet at right
angles, the moving frame is drawn with axes that meet at an acute
angle. The frames are actually equivalent. The asymmetry is due to
unavoidable distortions in how spacetime coordinates can map onto a
Cartesian plane, and should be considered no stranger than the manner
in which, on a
Figure 2-4. The light cone centered on an event divides the rest of spacetime into the future, the past, and "elsewhere". In Fig. 2-4, event O is at the origin of a spacetime diagram, and the two diagonal lines represent all events that have zero spacetime interval with respect to the origin event. These two lines form what is called the light cone of the event O, since adding a second spatial dimension (Fig. 2‑5) makes the appearance that of two right circular cones meeting with their apices at O. One cone extends into the future (t>0), the other into the past (t<0). Figure 2-5.
A light (double) cone divides spacetime into separate regions with respect to its apex. The interior of the future light cone consists of all events that are separated from the apex by more time (temporal distance) than necessary to cross their spatial distance at lightspeed; these events comprise the timelike future of the event O. Likewise, the timelike past comprises the interior events of the past light cone. So in timelike intervals Δct is greater than Δx, making timelike intervals positive. The region exterior to the light cone consists of events that are separated from the event O by more space than can be crossed at lightspeed in the given time. These events comprise the so-called spacelike region of the event O, denoted "Elsewhere" in Fig. 2‑4. Events on the light cone itself are said to be lightlike (or null separated) from O. Because of the invariance of the spacetime interval, all observers will assign the same light cone to any given event, and thus will agree on this division of spacetime.[18]:220 The light cone has an essential role within the concept of causality. It is possible for a not-faster-than-light-speed signal to travel from the position and time of O to the position and time of D (Fig. 2‑4). It is hence possible for event O to have a causal influence on event D. The future light cone contains all the events that could be causally influenced by O. Likewise, it is possible for a not-faster-than-light-speed signal to travel from the position and time of A, to the position and time of O. The past light cone contains all the events that could have a causal influence on O. In contrast, assuming that signals cannot travel faster than the speed of light, any event, like e.g. B or C, in the spacelike region (Elsewhere), cannot either effect event O, nor can they be affected by event O employing such signalling. Under this assumption any causal relationship between event O and any events in the spacelike region of a light cone is excluded.[26] Return to Introduction Relativity of simultaneity[edit] Click here for a brief section summary Figure 2-6. Animation illustrating relativity of simultaneity. All observers will agree that for any given event, an event within the given event's future light cone occurs after the given event. Likewise, for any given event, an event within the given event's past light cone occurs before the given event. The before-after relationship observed for timelike-separated events remains unchanged no matter what the reference frame of the observer, i.e. no matter how the observer may be moving. The situation is quite different for spacelike-separated events. Fig. 2‑4 was drawn from the reference frame of an observer moving at v = 0. From this reference frame, event C is observed to occur after event O, and event B is observed to occur before event O. From a different reference frame, the orderings of these non-causally-related events can be reversed. In particular, one notes that if two events are simultaneous in a particular reference frame, they are necessarily separated by a spacelike interval and thus are noncausally related. The observation that simultaneity is not absolute, but depends on the observer's reference frame, is termed the relativity of simultaneity.[27] Fig. 2-6 illustrates the use of spacetime diagrams in the analysis of the relativity of simultaneity. The events in spacetime are invariant, but the coordinate frames transform as discussed above for Fig. 2‑3. The three events (A, B, C) are simultaneous from the reference frame of an observer moving at v = 0. From the reference frame of an observer moving at v = 0.3 c, the events appear to occur in the order C, B, A. From the reference frame of an observer moving at v = −0.5 c, the events appear to occur in the order A, B, C. The white line represents a plane of simultaneity being moved from the past of the observer to the future of the observer, highlighting events residing on it. The gray area is the light cone of the observer, which remains invariant. A spacelike spacetime interval gives the same distance that an observer would measure if the events being measured were simultaneous to the observer. A spacelike spacetime interval hence provides a measure of proper distance, i.e. the true distance = − s 2 . textstyle sqrt -s^ 2 . Likewise, a timelike spacetime interval gives the same measure of time as would be presented by the cumulative ticking of a clock that moves along a given world line. A timelike spacetime interval hence provides a measure of the proper time = s 2 textstyle sqrt s^ 2 .[18]:220–221 Return to Introduction Invariant hyperbola[edit] Click here for a brief section summary Figure 2-7. (a) Families of invariant hyperbolae. (b) Hyperboloids of two sheets and one sheet. In
( c t ) 2 − x 2 = ± s 2 displaystyle (ct)^ 2 -x^ 2 =pm s^ 2 ; with s 2 displaystyle s^ 2 ; some positive real constant. These equations describe two families of hyperbolae in an x–ct spacetime diagram, which are termed invariant hyperbolae. In Fig. 2‑7a, each magenta hyperbola connects all events having some fixed spacelike separation from the origin, while the green hyperbolae connect events of equal timelike separation. Fig. 2‑7b reflects the situation in (1+2)-dimensional Minkowski spacetime (one temporal and two spatial dimensions) with the corresponding hyperboloids. Each timelike interval generates a hyperboloid of one sheet, while each spacelike interval generates a hyperboloid of two sheets. The (1+2)-dimensional boundary between space- and timelike hyperboloids, established by the events forming a zero spacetime interval to the origin, is made up by degenerating the hyperboloids to the light cone. In (1+1)-dimensions the hyperbolae degenerate to the two grey 45°-lines depicted in Fig. 2‑7a. Note on nomenclature: The magenta hyperbolae, which cross the x axis, are termed timelike (in contrast to spacelike) hyperbolae because all "distances" to the origin along the hyperbola are timelike intervals. Because of that, these hyperbolae represent actual paths that can be traversed by (constantly accelerating) particles in spacetime: between any two events on one hyperbola a causality relation is possible, because the inverse of the slope –representing the necessary speed– for all secants is less than c displaystyle c . On the other hand, the green hyperbolae, which cross the ct axis, are termed spacelike, because all intervals along these hyperbolae are spacelike intervals: no causality is possible between any two points on one of these hyperbolae, because all secants represent speeds larger than c displaystyle c .
Return to Introduction
Figure 2-8. The invariant hyperbola comprises the points that can be reached from the origin in a fixed proper time by clocks traveling at different speeds. Fig. 2-8 illustrates the invariant hyperbola for all events that can be reached from the origin in a proper time of 5 meters (approximately 6992167000000000000♠1.67×10−8 s). Different world lines represent clocks moving at different speeds. A clock that is stationary with respect to the observer has a world line that is vertical, and the elapsed time measured by the observer is the same as the proper time. For a clock traveling at 0.3c, the elapsed time measured by the observer is 5.24 meters (6992175000000000000♠1.75×10−8 s), while for a clock traveling at 0.7c, the elapsed time measured by the observer is 7.00 meters (6992233999999999999♠2.34×10−8 s). This illustrates the phenomenon known as time dilation. Clocks that travel faster take longer (in the observer frame) to tick out the same amount of proper time, and they travel further along the x–axis than they would have without time dilation.[18]:220–221 The measurement of time dilation by two observers in different inertial reference frames is mutual. If observer O measures the clocks of observer O′ as running slower in his frame, observer O′ in turn will measure the clocks of observer O as running slower. Figure 2-9. In this spacetime diagram, the 1 m length of the moving rod, as measured in the primed frame, is the foreshortened distance OC when projected onto the unprimed frame. Length contraction, like time dilation, is a manifestation of the relativity of simultaneity. Measurement of length requires measurement of the spacetime interval between two events that are simultaneous in one's frame of reference. But events that are simultaneous in one frame of reference are, in general, not simultaneous in other frames of reference. Fig. 2-9 illustrates the motions of a 1 m rod that is traveling at 0.5 c along the x axis. The edges of the blue band represent the world lines of the rod's two endpoints. The invariant hyperbola illustrates events separated from the origin by a spacelike interval of 1 m. The endpoints O and B measured when t′ = 0 are simultaneous events in the S′ frame. But to an observer in frame S, events O and B are not simultaneous. To measure length, the observer in frame S measures the endpoints of the rod as projected onto the x-axis along their world lines. The projection of the rod's world sheet onto the x axis yields the foreshortened length OC.[3]:125 (not illustrated) Drawing a vertical line through A so that it intersects the x' axis demonstrates that, even as OB is foreshortened from the point of view of observer O, OA is likewise foreshortened from the point of view of observer O′. In the same way that each observer measures the other's clocks as running slow, each observer measures the other's rulers as being contracted. Return to Introduction Mutual time dilation and the twin paradox[edit] Main article: Twin paradox Mutual time dilation[edit] Click here for a brief section summary Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts. The worry is that if observer A measures observer B's clocks as running slowly, simply because B is moving at speed v relative to A, then the principle of relativity requires that observer B likewise measures A's clocks as running slowly. This is an important question that "goes to the heart of understanding special relativity."[18]:198 Basically, A and B are performing two different measurements. In order to measure the rate of ticking of one of B's clocks, A must use two of his own clocks, the first to record the time where B's clock first ticked at the first location of B, and second to record the time where B's clock emitted its second tick at the next location of B. Observer A needs two clocks because B is moving, so a grand total of three clocks are involved in the measurement. A's two clocks must be synchronized in A's frame. Conversely, B requires two clocks synchronized in her frame to record the ticks of A's clocks at the locations where A's clocks emitted their ticks. Therefore, A and B are performing their measurements with different sets of three clocks each. Since they are not doing the same measurement with the same clocks, there is no inherent necessity that the measurements be reciprocally "consistent" such that, if one observer measures the other's clock to be slow, the other observer measures the one's clock to be fast.[18]:198–199 In regards to mutual length contraction, Fig. 2‑9 illustrates that the primed and unprimed frames are mutually rotated by a hyperbolic angle (analogous to ordinary angles in Euclidean geometry).[note 6] Because of this rotation, the projection of a primed meter-stick onto the unprimed x-axis is foreshortened, while the projection of an unprimed meter-stick onto the primed x′-axis is likewise foreshortened. Figure 2-10. Mutual time dilation Fig. 2-10 reinforces previous discussions about mutual time dilation.
In this figure, Events A and C are separated from event O by equal
timelike intervals. From the unprimed frame, events A and B are
measured as simultaneous, but more time has passed for the unprimed
observer than has passed for the primed observer. From the primed
frame, events C and D are measured as simultaneous, but more time has
passed for the primed observer than has passed for the unprimed
observer. Each observer measures the clocks of the other observer as
running more slowly.[3]:124
Please note the importance of the word "measure". An observer's state
of motion cannot affect an observed object, but it can affect the
observer's observations of the object.
In Fig. 2-10, each line drawn parallel to the x axis represents a line
of simultaneity for the unprimed observer. All events on that line
have the same time value of ct. Likewise, each line drawn parallel to
the x′ axis represents a line of simultaneity for the primed
observer. All events on that line have the same time value of ct′.
Return to Introduction
Twin paradox[edit]
Click here for a brief section summary
Elementary introductions to special relativity often illustrate the
differences between Galilean relativity and special relativity by
posing a series of supposed "paradoxes". All paradoxes are, in
reality, merely ill-posed or misunderstood problems, resulting from
our unfamiliarity with velocities comparable to the speed of light.
The remedy is to solve many problems in special relativity and to
become familiar with its so-called counter-intuitive predictions. The
geometrical approach to studying spacetime is considered one of the
best methods for developing a modern intuition.[28]
The twin paradox is a thought experiment involving identical twins,
one of whom makes a journey into space in a high-speed rocket,
returning home to find that the twin who remained on Earth has aged
more. This result appears puzzling because each twin observes the
other twin as moving, and so at first glance, it would appear that
each should find the other to have aged less. The twin paradox
sidesteps the justification for mutual time dilation presented above
by avoiding the requirement for a third clock.[18]:207 Nevertheless,
the twin paradox is not a true paradox because it is easily understood
within the context of special relativity.
The impression that a paradox exists stems from a misunderstanding of
what special relativity states.
Figure 2-11.
Deeper analysis is needed before we can understand why these
distinctions should result in a difference in the twins' ages.
Consider the spacetime diagram of Fig. 2‑11. This presents the
simple case of a twin going straight out along the x axis and
immediately turning back. From the standpoint of the stay-at-home
twin, there is nothing puzzling about the twin paradox at all. The
proper time measured along the traveling twin's world line from O to
C, plus the proper time measured from C to B, is less than the
stay-at-home twin's proper time measured from O to A to B. More
complex trajectories require integrating the proper time between the
respective events along the curve (i.e. the path integral) to
calculate the total amount of proper time experienced by the traveling
twin.[29]
Complications arise if the twin paradox is analyzed from the traveling
twin's point of view.
For the rest of this discussion, we adopt Weiss's nomenclature,
designating the stay-at-home twin as Terence and the traveling twin as
Stella.[29]
We had previously noted that Stella is not in an inertial frame. Given
this fact, it is sometimes stated that full resolution of the twin
paradox requires general relativity. This is not true.[29]
A pure SR analysis would be as follows: Analyzed in Stella's rest
frame, she is motionless for the entire trip. When she fires her
rockets for the turnaround, she experiences a pseudo force which
resembles a gravitational force.[29] Figs. 2‑6 and 2‑11
illustrate the concept of lines (planes) of simultaneity: Lines
parallel to the observer's x-axis (xy-plane) represent sets of events
that are simultaneous in the observer frame. In Fig. 2‑11, the
blue lines connect events on Terence's world line which, from Stella's
point of view, are simultaneous with events on her world line.
(Terence, in turn, would observe a set of horizontal lines of
simultaneity.) Throughout both the outbound and the inbound legs of
Stella's journey, she measures Terence's clocks as running slower than
her own. But during the turnaround (i.e. between the bold blue lines
in the figure), a shift takes place in the angle of her lines of
simultaneity, corresponding to a rapid skip-over of the events in
Terence's world line that Stella considers to be simultaneous with her
own. Therefore, at the end of her trip, Stella finds that Terence has
aged more than she has.[29]
Although general relativity is not required to analyze the twin
paradox, application of the
Galilean transformations[edit] Main article: Galilean group Click here for a brief section summary A basic goal is to be able to compare measurements made by observers in relative motion. Say we have an observer O in frame S who has measured the time and space coordinates of an event, assigning this event three Cartesian coordinates and the time as measured on his lattice of synchronized clocks (x, y, z, t) (see Fig. 1‑1). A second observer O′ in a different frame S′ measures the same event in her coordinate system and her lattice of synchronized clocks (x′, y′, z′, t′). Since we are dealing with inertial frames, neither observer is under acceleration, and a simple set of equations allows us to relate coordinates (x, y, z, t) to (x′, y′, z′, t′). Given that the two coordinate systems are in standard configuration, meaning that they are aligned with parallel (x, y, z) coordinates and that t = 0 when t′ = 0, the coordinate transformation is as follows:[30][31] x ′ = x − v t displaystyle x'=x-vt y ′ = y displaystyle y'=y z ′ = z displaystyle z'=z t ′ = t . displaystyle t'=t. Figure 3-1. Galilean
Fig. 3-1 illustrates that in Newton's theory, time is universal, not the velocity of light.[32]:36–37 Consider the following thought experiment: The red arrow illustrates a train that is moving at 0.4 c with respect to the platform. Within the train, a passenger shoots a bullet with a speed of 0.4 c in the frame of the train. The blue arrow illustrates that a person standing on the train tracks measures the bullet as traveling at 0.8 c. This is in accordance with our naive expectations. More generally, assume that frame S′ is moving at velocity v with respect to frame S. Within frame S′, observer O′ measures an object moving with velocity u′. What is its velocity u with respect to frame S? Since x = ut, x′ = x − vt, and t = t′, we can write x′ = ut − vt = (u − v)t = (u − v)t′. This leads to u′ = x′/t′ and ultimately u ′ = u − v displaystyle u'=u-v or u = u ′ + v , displaystyle u=u'+v, which is the common-sense Galilean law for the addition of velocities.
Return to Introduction
Relativistic composition of velocities[edit]
Main article:
Figure 3-2. Relativistic composition of velocities The composition of velocities is quite different in relativistic spacetime. To reduce the complexity of the equations slightly, we introduce a common shorthand for the ratio of the speed of an object relative to light, β = v / c displaystyle beta =v/c Fig. 3-2a illustrates a red train that is moving forward at a speed given by v/c = β = s/a. From the primed frame of the train, a passenger shoots a bullet with a speed given by u′/c = β′ = n/m, where the distance is measured along a line parallel to the red x′ axis rather than parallel to the black x axis. What is the composite velocity u of the bullet relative to the platform, as represented by the blue arrow? Referring to Fig. 3‑2b: From the platform, the composite speed of the bullet is given by u = c(s + r)/(a + b). The two yellow triangles are similar because they are right triangles that share a common angle α. In the large yellow triangle, the ratio s/a = v/c = β. The ratios of corresponding sides of the two yellow triangles are constant, so that r/a = b/s = n/m = β′. So b = u′s/c and r = u′a/c. Substitute the expressions for b and r into the expression for u in step 1 to yield Einstein's formula for the addition of velocities:[32]:42–48 u = v + u ′ 1 + ( v u ′ / c 2 ) . displaystyle u= v+u' over 1+(vu'/c^ 2 ) . The relativistic formula for addition of velocities presented above exhibits several important features: If u′ and v are both very small compared with the speed of light, then the product vu′/c2 becomes vanishingly small, and the overall result becomes indistinguishable from the Galilean formula (Newton's formula) for the addition of velocities: u = u′ + v. The Galilean formula is a special case of the relativistic formula applicable to low velocities. If u′ is set equal to c, then the formula yields u = c regardless of the starting value of v. The velocity of light is the same for all observers regardless their motions relative to the emitting source.[32]:49 Return to Introduction
Figure 3-3.
Click here for a brief section summary We had previously discussed, in qualitative terms, time dilation and length contraction. It is straightforward to obtain quantitative expressions for these effects. Fig. 3‑3 is a composite image containing individual frames taken from two previous animations, simplified and relabeled for the purposes of this section. To reduce the complexity of the equations slightly, we see in the literature a variety of different shorthand notations for ct : T = c t displaystyle mathrm T =ct and w = c t displaystyle w=ct are common. One also sees very frequently the use of the convention c = 1. displaystyle c=1. Figure 3-4. Lorentz factor as a function of velocity In Fig. 3-3a, segments OA and OK represent equal spacetime intervals.
O B = O K / 1 − v 2 / c 2 . textstyle OB=OK/ sqrt 1-v^ 2 /c^ 2 . The expression involving the square root symbol appears very frequently in relativity, and one over the expression is called the Lorentz factor, denoted by the Greek letter gamma γ displaystyle gamma :[33] γ = 1 1 − v 2 / c 2 = 1 1 − β 2 displaystyle gamma = frac 1 sqrt 1-v^ 2 /c^ 2 = frac 1 sqrt 1-beta ^ 2 We note that if v is greater than or equal to c, the expression for γ displaystyle gamma becomes physically meaningless, implying that c is the maximum
possible speed in nature. Next, we note that for any v greater than
zero, the Lorentz factor will be greater than one, although the shape
of the curve is such that for low speeds, the Lorentz factor is
extremely close to one.
In Fig. 3-3b, segments OA and OK represent equal spacetime intervals.
O B / O K = γ ( 1 − β 2 ) = 1 γ displaystyle OB/OK=gamma (1-beta ^ 2 )= frac 1 gamma Return to Introduction
Lorentz transformations[edit]
Main articles:
t ′ = γ ( t − v x c 2 ) x ′ = γ ( x − v t ) y ′ = y z ′ = z displaystyle begin aligned t'&=gamma left(t- frac vx c^ 2 right)\x'&=gamma left(x-vtright)\y'&=y\z'&=zend aligned The inverse Lorentz transformations are: t = γ ( t ′ + v x ′ c 2 ) x = γ ( x ′ + v t ′ ) y = y ′ z = z ′ displaystyle begin aligned t&=gamma left(t'+ frac vx' c^ 2 right)\x&=gamma left(x'+vt'right)\y&=y'\z&=z'end aligned When v ≪ c and x is small enough, the v2/c2 and vx/c2 terms approach zero, and the Lorentz transformations approximate to the Galilean transformations. As noted before, when we write t ′ = γ ( t − v x / c 2 ) , displaystyle t'=gamma (t-vx/c^ 2 ), x ′ = γ ( x − v t ) displaystyle x'=gamma (x-vt) and so forth, we most often really mean Δ t ′ = γ ( Δ t − v Δ x / c 2 ) , displaystyle Delta t'=gamma (Delta t-vDelta x/c^ 2 ), Δ x ′ = γ ( Δ x − v Δ t ) displaystyle Delta x'=gamma (Delta x-vDelta t) and so forth. Although, for brevity, we write the Lorentz transformation equations without deltas, it should be understood that x means Δx, etc. We are, in general, always concerned with the space and time differences between events. Note on nomenclature: Calling one set of transformations the normal Lorentz transformations and the other the inverse transformations is misleading, since there is no intrinsic difference between the frames. Different authors call one or the other set of transformations the "inverse" set. The forwards and inverse transformations are trivially related to each other, since the S frame can only be moving forwards or reverse with respect to S′. So inverting the equations simply entails switching the primed and unprimed variables and replacing v with −v.[34]:71–79 Example: Terence and Stella are at an Earth-to-Mars space race. Terence is an official at the starting line, while Stella is a participant. At time t = t′ = 0, Stella's spaceship accelerates instantaneously to a speed of 0.5 c. The distance from Earth to Mars is 300 light-seconds (about 7010900000000000000♠90.0×106 km). Terence observes Stella crossing the finish-line clock at t = 600.00 s. But Stella observes the time on her ship chronometer to be t′ = γ displaystyle gamma (t − vx/c2) = 519.62 s as she passes the finish line, and she calculates the distance between the starting and finish lines, as measured in her frame, to be 259.81 light-seconds (about 7010779000000000000♠77.9×106 km). 1). Return to Introduction Deriving the Lorentz transformations[edit] Main article: Derivations of the Lorentz transformations Figure 3-5. Derivation of Lorentz Transformation There have been many dozens of derivations of the Lorentz
transformations since Einstein's original work in 1905, each with its
particular focus. Although Einstein's derivation was based on the
invariance of the speed of light, there are other physical principles
that may serve as starting points. Ultimately, these alternative
starting points can be considered different expressions of the
underlying principle of locality, which states that the influence that
one particle exerts on another can not be transmitted
instantaneously.[35]
The derivation given here and illustrated in Fig. 3‑5 is based
on one presented by Bais[32]:64–66 and makes use of previous results
from the Relativistic Composition of Velocities,
We start by noting that there can be no such thing as length
expansion/contraction in the transverse directions. y' must equal y
and z′ must equal z, otherwise whether a fast moving 1 m ball
could fit through a 1 m circular hole would depend on the
observer. The first postulate of relativity states that all inertial
frames are equivalent, and transverse expansion/contraction would
violate this law.[34]:27–28
From the drawing, w = a + b and x = r + s
From previous results using similar triangles, we know that
s/a = b/r = v/c = β.
We know that because of time dilation, a = γw′
Substituting equation (4) into s/a = β yields
s = γw′β.
w = γ w ′ + β γ x ′ displaystyle w=gamma w'+beta gamma x' x = γ x ′ + β γ w ′ displaystyle x=gamma x'+beta gamma w' The above equations are alternate expressions for the t and x
equations of the inverse Lorentz transformation, as can be seen by
substituting ct for w, ct′ for w′, and v/c for β. From the
inverse transformation, the equations of the forwards transformation
can be derived by solving for t′ and x′.
Return to Introduction
Linearity of the Lorentz transformations[edit]
The Lorentz transformations have a mathematical property called
linearity, since x' and t' are obtained as linear combinations of x
and t, with no higher powers involved. The linearity of the
transformation reflects a fundamental property of spacetime that we
tacitly assumed while performing the derivation, namely, that the
properties of inertial frames of reference are independent of location
and time. In the absence of gravity, spacetime looks the same
everywhere.[32]:67 All inertial observers will agree on what
constitutes accelerating and non-accelerating motion.[34]:72–73 Any
one observer can use her own measurements of space and time, but there
is nothing absolute about them. Another observer's conventions will do
just as well.[18]:190
A result of linearity is that if two Lorentz transformations are
applied sequentially, the result is also a Lorentz transformation.
Example: Terence observes Stella speeding away from him at
0.500 c, and he can use the Lorentz transformations with
β = 0.500 to relate Stella's measurements to his own.
Stella, in her frame, observes Ursula traveling away from her at
0.250 c, and she can use the Lorentz transformations with
β = 0.250 to relate Ursula's measurements with her own.
Because of the linearity of the transformations and the relativistic
composition of velocities, Terence can use the Lorentz transformations
with β = 0.666 to relate Ursula's measurements with his
own.
Return to Introduction
Doppler effect[edit]
Main articles:
f = 1 1 + β s f 0 displaystyle f= frac 1 1+beta _ s f_ 0 On the other hand, given the scenario where source is stationary, and the receiver is moving directly away from the source at a speed of vr for a velocity parameter of βr, the wavelength is not changed, but the transmission velocity of the waves relative to the receiver is decreased, and the observed frequency f is given by f = ( 1 − β r ) f 0 displaystyle f=(1-beta _ r )f_ 0 Figure 3-6.
Light, unlike sound or water ripples, does not propagate through a
medium, and there is no distinction between a source moving away from
the receiver or a receiver moving away from the source.
Fig. 3‑6 illustrates a relativistic spacetime diagram showing a
source separating from the receiver with a velocity parameter β, so
that the separation between source and receiver at time w is βw.
Because of time dilation, w = γw'. Since the slope of the
green light ray is −1, T = w+βw = γw'(1+β). Hence, the
relativistic
f = 1 − β 1 + β f 0 . displaystyle f= sqrt frac 1-beta 1+beta ,f_ 0 . Return to Introduction Transverse Doppler effect[edit] Figure 3-7. Transverse
Suppose that a source, moving in a straight line, is at its closest point to the receiver. It would appear that the classical analysis predicts that the receiver detects no Doppler shift. Due to subtleties in the analysis, that expectation is not necessarily true. Nevertheless, when appropriately defined, transverse Doppler shift is a relativistic effect that has no classical analog. The subtleties are these:[34]:94–96 Fig. 3-7a. If a source, moving in a straight line, is crossing the receiver's field of view, what is the frequency measurement when the source is at its closest approach to the receiver? Fig. 3-7b. If a source is moving in a straight line, what is the frequency measurement when the receiver sees the source as being closest to it? Fig. 3-7c. If receiver is moving in a circle around the source, what frequency does the receiver measure? Fig. 3-7d. If the source is moving in a circle around the receiver, what frequency does the receiver measure? In scenario (a), when the source is closest to the receiver, the light hitting the receiver actually comes from a direction where the source had been some time back, and it has a significant longitudinal component, making an analysis from the frame of the receiver tricky. It is easier to make the analysis from S', the frame of the source. The point of closest approach is frame-independent and represents the moment where there is no change in distance versus time (i.e. dr/dt = 0 where r is the distance between receiver and source) and hence no longitudinal Doppler shift. The source observes the receiver as being illuminated by light of frequency f', but also observes the receiver as having a time-dilated clock. In frame S, the receiver is therefore illuminated by blueshifted light of frequency f = f ′ γ = f ′ / 1 − β 2 displaystyle f=f'gamma =f'/ sqrt 1-beta ^ 2 Scenario (b) is best analyzed from S, the frame of the receiver. The illustration shows the receiver being illuminated by light from when the source was closest to the receiver, even though the source has moved on. Because the source's clocks are time dilated, and since dr/dt was equal to zero at this point, the light from the source, emitted from this closest point, is redshifted with frequency f = f ′ / γ = f ′ 1 − β 2 displaystyle f=f'/gamma =f' sqrt 1-beta ^ 2 Scenarios (c) and (d) can be analyzed by simple time dilation arguments. In (c), the receiver observes light from the source as being blueshifted by a factor of γ displaystyle gamma , and in (d), the light is redshifted. The only seeming complication is that the orbiting objects are in accelerated motion. However, if an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation. (The converse, however, is not true.)[34]:94–96 Most reports of transverse Doppler shift refer to the effect as a redshift and analyze the effect in terms of scenarios (b) or (d).[note 7] Return to Introduction Energy and momentum[edit] Main articles: Four-momentum, Momentum, and Mass–energy equivalence Click here for a brief section summary Extending momentum to four dimensions[edit] Figure 3-8. Relativistic spacetime momentum vector In classical mechanics, the state of motion of a particle is
characterized by its mass and its velocity. Linear momentum, the
product of a particle's mass and velocity, is a vector quantity,
possessing the same direction as the velocity: p = mv. It is
a conserved quantity, meaning that if a closed system is not affected
by external forces, its total linear momentum cannot change.
In relativistic mechanics, the momentum vector is extended to four
dimensions. Added to the momentum vector is a time component that
allows the spacetime momentum vector to transform like the spacetime
position vector (x, t). In exploring the properties of the
spacetime momentum, we start, in Fig. 3‑8a, by examining what a
particle looks like at rest. In the rest frame, the spatial component
of the momentum is zero, i.e. p = 0, but the time component
equals mc.
We can obtain the transformed components of this vector in the moving
frame by using the Lorentz transformations, or we can read it directly
from the figure because we know that (mc)' = γmc and
p' = −βγmc, since the red axes are rescaled by gamma.
Fig. 3‑8b illustrates the situation as it appears in the moving
frame. It is apparent that the space and time components of the
four-momentum go to infinity as the velocity of the moving frame
approaches c.[32]:84–87
We will use this information shortly to obtain an expression for the
four-momentum.
Return to Introduction
Figure 3-9. Energy and momentum of light in different inertial frames Light particles, or photons, travel at the speed of c, the constant
that is conventionally known as the speed of light. This statement is
not a tautology, since many modern formulations of relativity do not
start with constant speed of light as a postulate. Photons therefore
propagate along a light-like world line and, in appropriate units,
have equal space and time components for every observer.
A consequence of
In the low speed limit as β = v/c approaches zero, γ displaystyle gamma approaches 1, so the spatial component of the relativistic momentum βγmc = γmv approaches mv, the classical term for momentum. Following this perspective, γm can be interpreted as a relativistic generalization of m. Einstein proposed that the relativistic mass of an object increases with velocity according to the formula mrel = γm. Likewise, comparing the time component of the relativistic momentum with that of the photon, γmc = mrelc = E/c, so that Einstein arrived at the relationship E = mrelc2. Simplified to the case of zero velocity, this is Einstein's famous equation relating energy and mass. Another way of looking at the relationship between mass and energy is to consider a series expansion of γmc2 at low velocity: E = γ m c 2 = m c 2 1 − β 2 displaystyle E=gamma mc^ 2 = frac mc^ 2 sqrt 1-beta ^ 2 ≈ m c 2 + 1 2 m v 2 . . . displaystyle approx mc^ 2 + frac 1 2 mv^ 2 ... The second term is just an expression for the kinetic energy of the
particle.
E 2 − p 2 c 2 = m 2 c 4 displaystyle E^ 2 -p^ 2 c^ 2 =m^ 2 c^ 4 This formula applies to all particles, massless as well as massive. For massless photons, it yields the same relationship that we had earlier established, E = ±pc.[32]:90–92 Return to Introduction Four-momentum[edit] Because of the close relationship between mass and energy, the four-momentum (also called 4‑momentum) is also called the energy-momentum 4‑vector. Using an uppercase P to represent the four-momentum and a lowercase p to denote the spatial momentum, the four-momentum may be written as P ≡ ( E / c , p → ) = ( E / c , p x , p y , p z ) displaystyle Pequiv (E/c, vec p )=(E/c,p_ x ,p_ y ,p_ z ) or alternatively, P ≡ ( E , p → ) = ( E , p x , p y , p z ) displaystyle Pequiv (E, vec p )=(E,p_ x ,p_ y ,p_ z ) using the convention that c = 1. displaystyle c=1. [34]:129–130,180 Return to Introduction
Conservation laws[edit]
Main article: Conservation law
Click here for a brief section summary
In physics, conservation laws state that certain particular measurable
properties of an isolated physical system do not change as the system
evolves over time. In 1915,
Figure 3-10. Relativistic conservation of momentum To understand how the Newtonian view of conservation of momentum needs to be modified in a relativistic context, we examine the problem of two colliding bodies limited to a single dimension. In Newtonian mechanics, two extreme cases of this problem may be distinguished yielding mathematics of minimum complexity: (1) The two bodies rebound from each other in a completely elastic collision. (2) The two bodies stick together and continue moving as a single particle. This second case is the case of completely inelastic collision. For both cases (1) and (2), momentum, mass, and total energy are conserved. However, kinetic energy is not conserved in cases of inelastic collision. A certain fraction of the initial kinetic energy is converted to heat. In case (2), two masses with momentums p1 = m1v1 and p2 = m2v2 collide to produce a single particle of conserved mass m = m1 + m2 traveling at the center of mass velocity of the original system, vcm = (m1v1 + m2v2)/(m1 + m2). The total momentum p = p1 + p2 is conserved. Fig. 3‑10 illustrates the inelastic collision of two particles from a relativistic perspective. The time components E1/c and E2/c add up to total E/c of the resultant vector, meaning that energy is conserved. Likewise, the space components p1 and p2 add up to form p of the resultant vector. The four-momentum is, as expected, a conserved quantity. However, the invariant mass of the fused particle, given by the point where the invariant hyperbola of the total momentum intersects the energy axis, is not equal to the sum of the invariant masses of the individual particles that collided. Indeed, it is larger than the sum of the individual masses: m > m1 + m2.[32]:94–97 Looking at the events of this scenario in reverse sequence, we see that non-conservation of mass is a common occurrence: when an unstable elementary particle spontaneously decays into two lighter particles, total energy is conserved, but the mass is not. Part of the mass is converted into kinetic energy.[34]:134–138 Return to Introduction Choice of reference frames[edit] Figure 3-11.
(above) Lab Frame.
(right) Center of
The freedom to choose any frame in which to perform an analysis allows
us to pick one which may be particularly convenient. For analysis of
momentum and energy problems, the most convenient frame is usually the
"center-of-momentum frame" (also called the zero-momentum frame, or
COM frame). This is the frame in which the space component of the
system's total momentum is zero. Fig. 3‑11 illustrates the
breakup of a high speed particle into two daughter particles. In the
lab frame, the daughter particles are preferentially emitted in a
direction oriented along the original particle's trajectory. In the
COM frame, however, the two daughter particles are emitted in opposite
directions, although their masses and the magnitude of their
velocities are generally not the same.
Return to Introduction
Energy and momentum conservation[edit]
In a Newtonian analysis of interacting particles, transformation
between frames is simple because all that is necessary is to apply the
Figure 3-12a. Energy-momentum diagram for decay of a charged pion Figure 3-12b. Graphing calculator analysis of charged pion decay. The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass. Mass is no longer conserved independently, because it has been subsumed into the total relativistic energy. This makes the relativistic conservation of energy a simpler concept than in nonrelativistic mechanics, because the total energy is conserved without any qualifications. Kinetic energy converted into heat or internal potential energy shows up as an increase in mass.[34]:127 Example: Because of the equivalence of mass and energy, elementary particle masses are customarily stated in energy units, where 1 MeV = 1×106 electron volts. A charged pion is a particle of mass 139.57 MeV (approx. 273 times the electron mass). It is unstable, and decays into a muon of mass 105.66 MeV (approx. 207 times the electron mass) and an antineutrino, which has an almost negligible mass. The difference between the pion mass and the muon mass is 33.91 MeV. π− → μ− + ν μ Fig. 3‑12a illustrates the energy-momentum diagram for this decay reaction in the rest frame of the pion. Because of its negligible mass, a neutrino travels at very nearly the speed of light. The relativistic expression for its energy, like that of the photon, is Eν = pc, which is also the value of the space component of its momentum. To conserve momentum, the muon has the same value of the space component of the neutrino's momentum, but in the opposite direction. Algebraic analyses of the energetics of this decay reaction are available online,[39] so Fig. 3‑12b presents instead a graphing calculator solution. The energy of the neutrino is 29.79 MeV, and the energy of the muon is 33.91 − 29.79 = 4.12 MeV. Interestingly, most of the energy is carried off by the near-zero-mass neutrino. Return to Introduction Beyond the basics[edit] The topics in this section are of significantly greater technical difficulty than those in the preceding sections and are not essential for understanding Introduction to curved spacetime. Rapidity[edit] Main article: Rapidity Click here for a brief section summary Figure 4-1a. A ray through the unit circle x2 + y2 = 1 in the point (cos a, sin a), where a is twice the area between the ray, the circle, and the x-axis. Figure 4-1b. A ray through the unit hyperbola x2 - y2 = 1 in the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis. Figure 4-2. Plot of the three basic Hyperbolic functions: hyperbolic sine (sinh), hyperbolic cosine (cosh) and hyperbolic tangent (tanh). Sinh is red, cosh is blue and tanh is green. Lorentz transformations relate coordinates of events in one reference frame to those of another frame. Relativistic composition of velocities is used to add two velocities together. The formulas to perform the latter computations are nonlinear, making them more complex than the corresponding Galilean formulas. This nonlinearity is an artifact of our choice of parameters.[6]:47–59 We have previously noted that in an x–ct spacetime diagram, the points at some constant spacetime interval from the origin form an invariant hyperbola. We have also noted that the coordinate systems of two spacetime reference frames in standard configuration are hyperbolically rotated with respect to each other. The natural functions for expressing these relationships are the hyperbolic analogs of the trigonometric functions. Fig. 4‑1a shows a unit circle with sin(a) and cos(a), the only difference between this diagram and the familiar unit circle of elementary trigonometry being that a is interpreted, not as the angle between the ray and the x-axis, but as twice the area of the sector swept out by the ray from the x-axis. (Numerically, the angle and 2 × area measures for the unit circle are identical.) Fig. 4‑1b shows a unit hyperbola with sinh(a) and cosh(a), where a is likewise interpreted as twice the tinted area.[40] Fig. 4‑2 presents plots of the sinh, cosh, and tanh functions. For the unit circle, the slope of the ray is given by slope = tan a = sin a cos a . displaystyle text slope =tan a= frac sin a cos a . In the Cartesian plane, rotation of point (x, y) into point (x', y') by angle θ is given by ( x ′ y ′ ) = ( cos θ − sin θ sin θ cos θ ) ( x y ) . displaystyle begin pmatrix x'\y'\end pmatrix = begin pmatrix cos theta &-sin theta \sin theta &cos theta \end pmatrix begin pmatrix x\y\end pmatrix . In a spacetime diagram, the velocity parameter β displaystyle beta is the analog of slope. The rapidity, φ, is defined by[34]:96–99 β ≡ tanh ϕ ≡ v c , displaystyle beta equiv tanh phi equiv frac v c , where tanh ϕ = sinh ϕ cosh ϕ = e ϕ − e − ϕ e ϕ + e − ϕ . displaystyle tanh phi = frac sinh phi cosh phi = frac e^ phi -e^ -phi e^ phi +e^ -phi . The rapidity defined above is very useful in special relativity because many expressions take on a considerably simpler form when expressed in terms of it. For example, rapidity is simply additive in the collinear velocity-addition formula;[6]:47–59 β = β 1 + β 2 1 + β 1 β 2 = displaystyle beta = frac beta _ 1 +beta _ 2 1+beta _ 1 beta _ 2 = tanh ϕ 1 + tanh ϕ 2 1 + tanh ϕ 1 tanh ϕ 2 = displaystyle frac tanh phi _ 1 +tanh phi _ 2 1+tanh phi _ 1 tanh phi _ 2 = tanh ( ϕ 1 + ϕ 2 ) , displaystyle tanh(phi _ 1 +phi _ 2 ), or in other words, ϕ = ϕ 1 + ϕ 2 . displaystyle phi =phi _ 1 +phi _ 2 . The Lorentz transformations take a simple form when expressed in terms of rapidity. The γ factor can be written as γ = 1 1 − β 2 = 1 1 − tanh 2 ϕ displaystyle gamma = frac 1 sqrt 1-beta ^ 2 = frac 1 sqrt 1-tanh ^ 2 phi = cosh ϕ , displaystyle =cosh phi , γ β = β 1 − β 2 = tanh ϕ 1 − tanh 2 ϕ displaystyle gamma beta = frac beta sqrt 1-beta ^ 2 = frac tanh phi sqrt 1-tanh ^ 2 phi = sinh ϕ . displaystyle =sinh phi . Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called boosts. Substituting γ and γβ into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in the x direction may be written as ( c t ′ x ′ ) = ( cosh ϕ − sinh ϕ − sinh ϕ cosh ϕ ) ( c t x ) , displaystyle begin pmatrix ct'\x'end pmatrix = begin pmatrix cosh phi &-sinh phi \-sinh phi &cosh phi end pmatrix begin pmatrix ct\xend pmatrix , and the inverse Lorentz boost in the x direction may be written as ( c t x ) = ( cosh ϕ sinh ϕ sinh ϕ cosh ϕ ) ( c t ′ x ′ ) . displaystyle begin pmatrix ct\xend pmatrix = begin pmatrix cosh phi &sinh phi \sinh phi &cosh phi end pmatrix begin pmatrix ct'\x'end pmatrix . In other words, Lorentz boosts represent hyperbolic rotations in
Minkowski spacetime.[34]:96–99
The advantages of using hyperbolic functions are such that some
textbooks such as the classic ones by Taylor and Wheeler introduce
their use at a very early stage.[6][41] [note 8]
Return to Introduction
4‑vectors[edit]
Main article: Four-vector
Click here for a brief section summary
Four‑vectors have been mentioned above in context of the
energy-momentum 4‑vector, but without any great emphasis. Indeed,
none of the elementary derivations of special relativity require them.
But once understood, 4‑vectors, and more generally tensors, greatly
simplify the mathematics and conceptual understanding of special
relativity. Working exclusively with such objects leads to formulas
that are manifestly relativistically invariant, which is a
considerable advantage in non-trivial contexts. For instance,
demonstrating relativistic invariance of
A 0 ′ = γ ( A 0 − ( v / c ) A 1 ) A 1 ′ = γ ( A 1 − ( v / c ) A 0 ) A 2 ′ = A 2 A 3 ′ = A 3 displaystyle begin aligned A_ 0 '&=gamma left(A_ 0 -(v/c)A_ 1 right)\A_ 1 '&=gamma left(A_ 1 -(v/c)A_ 0 right)\A_ 2 '&=A_ 2 \A_ 3 '&=A_ 3 end aligned which comes from simply replacing ct with A0 and x with A1 in the earlier presentation of the Lorentz transformation. As usual, when we write x, t, etc. we generally mean Δx, Δt etc. The last three components of a 4–vector must be a standard vector in three-dimensional space. Therefore, a 4–vector must transform like (c Δt, Δx, Δy, Δz) under Lorentz transformations as well as rotations.[28]:36–59 Return to Introduction Properties of 4-vectors[edit] Closure under linear combination: If A and B are 4-vectors, then C = aA + aB is also a 4-vector. Inner-product invariance: If A and B are 4-vectors, then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a 3-vector. In the following, A → displaystyle vec A and B → displaystyle vec B are 3-vectors: A ⋅ B ≡ displaystyle Acdot Bequiv A 0 B 0 − A 1 B 1 − A 2 B 2 − A 3 B 3 ≡ displaystyle A_ 0 B_ 0 -A_ 1 B_ 1 -A_ 2 B_ 2 -A_ 3 B_ 3 equiv A 0 B 0 − A → ⋅ B → displaystyle A_ 0 B_ 0 - vec A cdot vec B In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in 3-space. Two vectors are said to be orthogonal if A ⋅ B = 0. displaystyle Acdot B=0. Unlike the case with 3-vectors, orthogonal 4-vectors are not necessarily at right angles with each other. The rule is that two 4-vectors are orthogonal if they are offset by equal and opposite angles from the 45° line which is the world line of a light ray. This implies that a lightlike 4-vector is orthogonal with itself. Invariance of the magnitude of a vector: The magnitude of a vector is the inner product of a 4-vector with itself, and is a frame-independent property. As with intervals, the magnitude may be positive, negative or zero, so that the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one for which A ⋅ A = 0 , displaystyle Acdot A=0, while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval c 2 t 2 − x 2 displaystyle c^ 2 t^ 2 -x^ 2 and the invariant length of the relativistic momentum vector E 2 − p 2 c 2 . displaystyle E^ 2 -p^ 2 c^ 2 . [34]:178–181[28]:36–59 Return to Introduction Examples of 4-vectors[edit] Displacement 4-vector: Otherwise known as the spacetime separation, this is (Δt, Δx, Δy, Δz), or for infinitesimal separations, (dt, dx, dy, dz). d S ≡ ( d t , d x , d y , d z ) displaystyle dSequiv (dt,dx,dy,dz)
d τ displaystyle dtau , where d τ displaystyle dtau is the proper time between the two events that yield dt, dx, dy, and dz. V ≡ d S d τ = ( d t , d x , d y , d z ) d t / γ = displaystyle Vequiv frac dS dtau = frac (dt,dx,dy,dz) dt/gamma = γ ( 1 , d x d t , d y d t , d z d t ) = displaystyle gamma left(1, frac dx dt , frac dy dt , frac dz dt right)= ( γ , γ v → ) displaystyle (gamma ,gamma vec v ) Figure 4-3a. The momentarily comoving reference frames of an accelerating particle as observed from a stationary frame. Figure 4-3b. The momentarily comoving reference frames along the trajectory of an accelerating observer (center). The 4-velocity is tangent to the world line of a particle, and has a length equal to one unit of time in the frame of the particle. An accelerated particle does not have an inertial frame in which it is always at rest. However, as stated before in the earlier discussion of the transverse Doppler effect, an inertial frame can always be found which is momentarily comoving with the particle. This frame, the momentarily comoving reference frame (MCRF), enables application of special relativity to the analysis of accelerated particles. Since photons move on null lines, d τ = 0 displaystyle dtau =0 for a photon, and a 4-velocity cannot be defined. There is no frame in which a photon is at rest, and no MCRF can be established along a photon's path. Energy-momentum 4-vector: As discussed in the section on Energy and momentum, P ≡ ( E / c , p → ) = ( E / c , p x , p y , p z ) displaystyle Pequiv (E/c, vec p )=(E/c,p_ x ,p_ y ,p_ z ) As indicated before, there are varying treatments for the energy-momentum 4-vector so that one may also see it expressed as ( E , p → ) displaystyle (E, vec p ) or ( E , p → c ) . displaystyle (E, vec p c). The first component is the total energy (including mass) of the particle (or system of particles) in a given frame, while the remaining components are its spatial momentum. The energy-momentum 4-vector is a conserved quantity. Acceleration 4-vector: This results from taking the derivative of the velocity 4-vector with respect to τ . displaystyle tau . A ≡ d V d τ = displaystyle Aequiv frac dV dtau = d d τ ( γ , γ v → ) = displaystyle frac d dtau (gamma ,gamma vec v )= γ ( d γ d t , d ( γ v → ) d t ) displaystyle gamma left( frac dgamma dt , frac d(gamma vec v ) dt right) Force 4-vector: This is the derivative of the momentum 4-vector with respect to τ . displaystyle tau . F ≡ d P d τ = displaystyle Fequiv frac dP dtau = γ ( d E d t , d p → d t ) = displaystyle gamma left( frac dE dt , frac d vec p dt right)= γ ( d E d t , f → ) displaystyle gamma left( frac dE dt , vec f right) As expected, the final components of the above 4-vectors are all standard 3-vectors corresponding to spatial 3-momentum, 3-force etc.[34]:178–181[28]:36–59 Return to Introduction 4-vectors and physical law[edit] The first postulate of special relativity declares the equivalency of all inertial frames. A physical law holding in one frame must apply in all frames, since otherwise it would be possible to differentiate between frames. As noted in the previous discussion of energy and momentum conservation, Newtonian momenta fail to behave properly under Lorentzian transformation, and Einstein preferred to change the definition of momentum to one involving 4-vectors rather than give up on conservation of momentum. Physical laws must be based on constructs that are frame independent. This means that physical laws may take the form of equations connecting scalars, which are always frame independent. However, equations involving 4-vectors require the use of tensors with appropriate rank, which themselves can be thought of as being built up from 4-vectors.[34]:186 Return to Introduction Acceleration[edit] Further information: Acceleration (special relativity) Click here for a brief section summary It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames. Actually, accelerating objects can generally be analyzed without needing to deal with accelerating frames at all. It is only when gravitation is significant that general relativity is required.[42] Properly handling accelerating frames does requires some care, however. The difference between special and general relativity is that (1) In special relativity, all velocities are relative, but acceleration is absolute. (2) In general relativity, all motion is relative, whether inertial, accelerating, or rotating. To accommodate this difference, general relativity uses curved spacetime.[42] In this section, we analyze several scenarios involving accelerated reference frames. Return to Introduction Dewan–Beran–Bell spaceship paradox[edit] Main article: Bell's spaceship paradox The Dewan–Beran–Bell spaceship paradox (Bell's spaceship paradox) is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues. Figure 4-4. Dewan–Beran–Bell spaceship paradox In Fig. 4‑4, two identical spaceships float in space and are at rest relative to each other. They are connected by a string which is capable of only a limited amount of stretching before breaking. At a given instant in our frame, the observer frame, both spaceships accelerate in the same direction along the line between them with the same constant proper acceleration.[note 9] Will the string break? The main article for this section recounts how, when the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution. Two lines of reasoning lead to opposite conclusions. Both arguments, which are presented below, are flawed even though one of them yields the correct answer.[34]:106,120–122 To observers in the rest frame, the spaceships start a distance L apart and remain the same distance apart during acceleration. During acceleration, L is a length contracted distance of the distance L' = γL in the frame of the accelerating spaceships. After a sufficiently long time, γ will increase to a sufficiently large factor that the string must break. Let A and B be the rear and front spaceships. In the frame of the spaceships, each spaceship sees the other spaceship doing the same thing that it is doing. A says that B has the same acceleration that he has, and B sees that A matches her every move. So the spaceships stay the same distance apart, and the string does not break.[34]:106,120–122 The problem with the first argument is that there is no "frame of the spaceships." There cannot be, because the two spaceships measure a growing distance between the two. Because there is no common frame of the spaceships, the length of the string is ill-defined. Nevertheless, the conclusion is correct, and the argument is mostly right. The second argument, however, completely ignores the relativity of simultaneity.[34]:106,120–122 Figure 4-5. The blue lines represent the world lines of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A' and B', the observers stop accelerating. The dotted line is a line of simultaneity for either observer after acceleration stops. A spacetime diagram (Fig. 4‑5) makes the correct solution to this paradox almost immediately evident. Two observers in Minkowski spacetime accelerate with constant magnitude k displaystyle k acceleration for proper time σ displaystyle sigma (acceleration and elapsed time measured by the observers themselves, not some inertial observer). They are comoving and inertial before and after this phase. In Minkowski geometry, the length of the spacelike line segment A ′ B ″ displaystyle A'B'' turns out to be greater than the length of the spacelike line segment A B displaystyle AB . The length increase can be calculated with the help of the Lorentz transformation. If, as illustrated in Fig. 4‑5, the acceleration is finished, the ships will remain at a constant offset in some frame S ′ . displaystyle S'. If x A displaystyle x_ A and x B = x A + L displaystyle x_ B =x_ A +L are the ships' positions in S , displaystyle S, the positions in frame S ′ displaystyle S' are:[43] x A ′ = γ ( x A − v t ) x B ′ = γ ( x A + L − v t ) L ′ = x B ′ − x A ′ = γ L displaystyle begin aligned x'_ A &=gamma left(x_ A -vtright)\x'_ B &=gamma left(x_ A +L-vtright)\L'&=x'_ B -x'_ A =gamma Lend aligned The "paradox", as it were, comes from the way that Bell constructed his example. In the usual discussion of Lorentz contraction, the rest length is fixed and the moving length shortens as measured in frame S displaystyle S . As shown in Fig. 4‑5, Bell's example asserts the moving lengths A B displaystyle AB and A ′ B ′ displaystyle A'B' measured in frame S displaystyle S to be fixed, thereby forcing the rest frame length A ′ B ″ displaystyle A'B'' in frame S ′ displaystyle S' to increase.
Return to Introduction
Accelerated observer with horizon[edit]
Main articles:
Figure 4-6. Accelerated relativistic observer with horizon. Another well-drawn illustration of the same topic may be viewed here. Fig. 4‑6 details various features of the traveler's motions with more specificity. At any given moment, her space axis is formed by a line passing through the origin and her current position on the hyperbola, while her time axis is the tangent to the hyperbola at her position. The velocity parameter β displaystyle beta approaches a limit of one as c t displaystyle ct increases. Likewise, γ displaystyle gamma approaches infinity. The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows: We remember that β = c t / x . displaystyle beta =ct/x. Since c 2 t 2 − x 2 = s 2 , displaystyle c^ 2 t^ 2 -x^ 2 =s^ 2 , we conclude that β ( c t ) = c t / c 2 t 2 − s 2 . displaystyle beta (ct)=ct/ sqrt c^ 2 t^ 2 -s^ 2 . γ = 1 / 1 − β 2 = displaystyle gamma =1/ sqrt 1-beta ^ 2 = c 2 t 2 − s 2 / s displaystyle sqrt c^ 2 t^ 2 -s^ 2 /s From the relativistic force law, F = d p / d t = displaystyle F=dp/dt= d p c / d ( c t ) = d ( β γ m c 2 ) / d ( c t ) . displaystyle dpc/d(ct)=d(beta gamma mc^ 2 )/d(ct). Substituting β ( c t ) displaystyle beta (ct) from step 2 and the expression for γ displaystyle gamma from step 3 yields F = m c 2 / s , displaystyle F=mc^ 2 /s, which is a constant expression.[32]:110–113 Fig. 4‑6 illustrates a specific calculated scenario. Terence
(A) and Stella (B) initially stand together 100 light hours from
the origin. Stella lifts off at time 0, her spacecraft accelerating at
0.01 c per hour. Every twenty hours, Terence radios updates to
Stella about the situation at home (solid green lines). Stella
receives these regular transmissions, but the increasing distance
(offset in part by time dilation) causes her to receive Terence's
communications later and later as measured on her clock, and she never
receives any communications from Terence after 100 hours on his clock
(dashed green lines).[32]:110–113
After 100 hours according to Terence's clock, Stella enters a dark
region. She has traveled outside Terence's timelike future. On the
other hand, Terence can continue to receive Stella's messages to him
indefinitely. He just has to wait long enough.
Basic propositions[edit]
Click here for a brief section summary
Newton's theories assumed that motion takes place against the backdrop
of a rigid Euclidean reference frame that extends throughout all space
and all time.
Figure 5-1. Tidal effects [Click here for additional details 1] In spacetime terms, the path of a satellite orbiting the Earth is not dictated by the distant influences of the Earth, Moon and Sun. Instead, the satellite moves through space only in response to local conditions. Since spacetime is everywhere locally flat when considered on a sufficiently small scale, the satellite is always following a straight line in its local inertial frame. We say that the satellite always follows along the path of a geodesic. No evidence of gravitation can be discovered following alongside the motions of a single particle.[6]:175–190 In any analysis of spacetime, evidence of gravitation requires that one observe the relative accelerations of two bodies or two separated particles. In Fig. 5‑1, two separated particles, free-falling in the gravitational field of the Earth, exhibit tidal accelerations due to local inhomogeneities in the gravitational field such that each particle follows a different path through spacetime. The tidal accelerations that these particles exhibit with respect to each other do not require forces for their explanation. Rather, Einstein described them in terms of the geometry of spacetime, i.e. the curvature of spacetime. These tidal accelerations are strictly local. It is the cumulative total effect of many local manifestations of curvature that result in the appearance of a gravitational force acting at a long range from Earth.[6]:175–190 Two central propositions underlie general relativity. The first crucial concept is coordinate independence: The laws of physics cannot depend on what coordinate system one uses. This is a major extension of the principle of relativity from the version used in special relativity, which states that the laws of physics must be the same for every observer moving in non-accelerated (inertial) reference frames. In general relativity, to use Einstein's own (translated) words, "the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion."[44]:113 This leads to an immediate issue: In accelerated frames, one feels forces that seemingly would enable one to assess one's state of acceleration in an absolute sense. Einstein resolved this problem through the principle of equivalence.[45]:137–149 Figure 5-2. Equivalence principle The equivalence principle states that in any sufficiently small region of space, the effects of gravitation are the same as those from acceleration. In Fig. 5-2, person A is in a spaceship, far from any massive objects, that undergoes a uniform acceleration of g. Person B is in a box resting on Earth. Provided that the spaceship is sufficiently small so that tidal effects are non-measurable (given the sensitivity of current gravity measurement instrumentation, A and B presumably should be Lilliputians), there are no experiments that A and B can perform which will enable them to tell which setting they are in.[45]:141–149 An alternative expression of the equivalence principle is to note that in Newton's universal law of gravitation, F = GMmg /r2 = mgg and in Newton's second law, F = m ia, there is no a priori reason why the gravitational mass mg should be equal to the inertial mass m i. The equivalence principle states that these two masses are identical.[45]:141–149 To go from the elementary description above of curved spacetime to a complete description of gravitation requires tensor calculus and differential geometry, topics both requiring considerable study. Without these mathematical tools, it is possible to write about general relativity, but it is not possible to demonstrate any non-trivial derivations. Rather than this section attempting to offer a (yet another) relatively non-mathematical presentation about general relativity, the reader is referred to the featured articles Introduction to general relativity and General relativity. Instead, the focus in this section will be to explore a handful of elementary scenarios that serve to give somewhat of the flavor of general relativity. Return to Introduction Curvature of time[edit] Click here for a brief section summary Figure 5-3. Einstein's argument suggesting gravitational redshift In the discussion of special relativity, forces played no more than a
background role.
E ′ E = h ν ′ h ν = displaystyle frac E' E = frac hnu ,' hnu = m m + m g h / c 2 = displaystyle frac m m+mgh/c^ 2 = 1 − g h c 2 displaystyle 1- frac gh c^ 2 A photon climbing in Earth's gravitational field loses energy and is
redshifted. Early attempts to measure this redshift through
astronomical observations were somewhat inconclusive, but definitive
laboratory observations were performed by Pound & Rebka (1959) and
later by Pound & Snider (1964).[46]
Light has an associated frequency, and this frequency may be used to
drive the workings of a clock. The gravitational redshift leads to an
important conclusion about time itself:
Δ s 2 = ( 1 − 2 G M c 2 r ) ( c Δ t ) 2 displaystyle Delta s^ 2 =left(1- frac 2GM c^ 2 r right)(cDelta t)^ 2 − ( Δ x ) 2 − ( Δ y ) 2 − ( Δ z ) 2 displaystyle -,(Delta x)^ 2 -(Delta y)^ 2 -(Delta z)^ 2 Return to Introduction Curvature of space[edit] Click here for a brief section summary The ( 1 − 2 G M / ( c 2 r ) ) displaystyle (1-2GM/(c^ 2 r)) coefficient in front of ( c Δ t ) 2 displaystyle (cDelta t)^ 2 describes the curvature of time in Newtonian gravitation, and this curvature completely accounts for all Newtonian gravitational effects. As expected, this correction factor is directly proportional to G displaystyle G and M displaystyle M , and because of the r displaystyle r in the denominator, the correction factor increases as one approaches the gravitating body, meaning that time is curved. But general relativity is a theory of curved space and curved time, so if there are terms modifying the spatial components of the spacetime interval presented above, shouldn't their effects be seen on, say, planetary and satellite orbits due to curvature correction factors applied to the spatial terms? The answer is that they are seen, but the effects are tiny. The reason is that planetary velocities are extremely small compared to the speed of light, so that for planets and satellites of the solar system, the ( c Δ t ) 2 displaystyle (cDelta t)^ 2 term dwarfs the spatial terms.[18]:234–238 Despite the minuteness of the spatial terms, the first indications that something was wrong with Newtonian gravitation were discovered over a century-and-a-half ago. In 1859, Urbain Le Verrier, in an analysis of available timed observations of transits of Mercury over the Sun's disk from 1697 to 1848, reported that known physics could not explain the orbit of Mercury, unless there possibly existed a planet or asteroid belt within the orbit of Mercury. The perihelion of Mercury's orbit exhibited an excess rate of precession over that which could be explained by the tugs of the other planets.[48] The ability to detect and accurately measure the minute value of this anomalous precession (only 43 arc seconds per tropical century) is testimony to the sophistication of 19th century astrometry. Figure 5-4.
As the famous astronomer who had earlier discovered the existence of Neptune "at the tip of his pen" by analyzing wobbles in the orbit of Uranus, Le Verrier's announcement triggered a two-decades long period of "Vulcan-mania", as professional and amateur astronomers alike hunted for the hypothetical new planet. This search included several false sightings of Vulcan. It was ultimately established that no such planet or asteroid belt existed.[49] In 1916, Einstein was to show that this anomalous precession of Mercury is explained by the spatial terms in the curvature of spacetime. Curvature in the temporal term, being simply an expression of Newtonian gravitation, has no part in explaining this anomalous precession. The success of his calculation was a powerful indication to Einstein's peers that the general theory of relativity could be correct. The most spectacular of Einstein's predictions was his calculation that the curvature terms in the spatial components of the spacetime interval could be measured in the bending of light around a massive body. Light has a slope of ±1 on a spacetime diagram. Its movement in space is equal to its movement in time. For the weak field expression of the invariant interval, Einstein calculated an exactly equal but opposite sign curvature in its spatial components.[18]:234–238 Δ s 2 = ( 1 − 2 G M c 2 r ) ( c Δ t ) 2 displaystyle Delta s^ 2 =left(1- frac 2GM c^ 2 r right)(cDelta t)^ 2 − ( 1 + 2 G M c 2 r ) [ ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 ] displaystyle -,left(1+ frac 2GM c^ 2 r right)left[(Delta x)^ 2 +(Delta y)^ 2 +(Delta z)^ 2 right] In Newton's gravitation, the ( 1 − 2 G M / ( c 2 r ) ) displaystyle (1-2GM/(c^ 2 r)) coefficient in front of ( c Δ t ) 2 displaystyle (cDelta t)^ 2 predicts bending of light around a star. In general relativity, the ( 1 + 2 G M / ( c 2 r ) ) displaystyle (1+2GM/(c^ 2 r)) coefficient in front of [ ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 ] displaystyle left[(Delta x)^ 2 +(Delta y)^ 2 +(Delta z)^ 2 right] predicts a doubling of the total bending.[18]:234–238 The story of the 1919 Eddington eclipse expedition and Einstein's rise to fame is well told elsewhere.[50] Return to Introduction Sources of spacetime curvature[edit] Click here for a brief section summary Figure 5-5. Contravariant components of the stress–energy tensor. In Newton's theory of gravitation, the only source of gravitational force is mass. In contrast, general relativity identifies several sources of spacetime curvature in addition to mass. In the Einstein field equations, the sources of gravity are presented on the right-hand side in T μ ν , displaystyle T_ mu nu , the stress–energy tensor. Fig. 5‑5 classifies the various sources of gravity in the stress-energy tensor: T 00 displaystyle T^ 00 (red): The total mass-energy density, including any contributions to the potential energy from forces between the particles, as well as kinetic energy from random thermal motions. T 0 i displaystyle T^ 0i and T i 0 displaystyle T^ i0 (orange): These are momentum density terms. Even if there is no bulk motion, energy may be transmitted by heat conduction, and the conducted energy will carry momentum. T i j displaystyle T^ ij are the rates of flow of the i-component of momentum per unit area in the j-direction. Even if there is no bulk motion, random thermal motions of the particles will give rise to momentum flow, so the i = j terms (green) represent isotropic pressure, and the i ≠ j terms (blue) represent shear stresses.[51] One important conclusion to be derived from the equations is that,
colloquially speaking, gravity itself creates gravity.[note 11] Energy
has mass. Even in Newtonian gravity, the gravitational field is
associated with an energy, E = mgh, called the gravitational potential
energy. In general relativity, the energy of the gravitational field
feeds back into creation of the gravitational field. This makes the
equations nonlinear and hard to solve in anything other than weak
field cases.[18]:240
Figure 5-6. (left) Mass-energy warps spacetime. (right) Rotating mass-energy distributions with angular momentum J generate gravitomagnetic fields H In special relativity, mass-energy is closely connected to momentum. As we have discussed earlier in the section on Energy and momentum, just as space and time are different aspects of a more comprehensive entity called spacetime, mass-energy and momentum are merely different aspects of a unified, four-dimensional quantity called four-momentum. In consequence, if mass-energy is a source of gravity, momentum must also be a source. The inclusion of momentum as a source of gravity leads to the prediction that moving or rotating masses can generate fields analogous to the magnetic fields generated by moving charges, a phenomenon known as gravitomagnetism.[52] Figure 5-7. Origin of gravitomagnetism. It is well known that the force of magnetism can be deduced by applying the rules of special relativity to moving charges. (An eloquent demonstration of this was presented by Feynman in volume II, chapter 13–6 of his Lectures on Physics, available online.[53]) Analogous logic can be used to demonstrate the origin of gravitomagnetism. In Fig. 5‑7a, two parallel, infinitely long streams of massive particles have equal and opposite velocities −v and +v relative to a test particle at rest and centered between the two. Because of the symmetry of the setup, the net force on the central particle is zero. Assume v << c so that velocities are simply additive. Fig. 5‑7b shows exactly the same setup, but in the frame of the upper stream. The test particle has a velocity of +v, and the bottom stream has a velocity of +2v. Since the physical situation has not changed, only the frame in which things are observed, the test particle should not be attracted towards either stream. But it is not at all clear that the forces exerted on the test particle are equal. (1) Since the bottom stream is moving faster than the top, each particle in the bottom stream has a larger mass energy than a particle in the top. (2) Because of Lorentz contraction, there are more particles per unit length in the bottom stream than in the top stream. (3) Another contribution to the active gravitational mass of the bottom stream comes from an additional pressure term which, at this point, we do not have sufficient background to discuss. All of these effects together would seemingly demand that the test particle be drawn towards the bottom stream. Figure 5-8. Relativistic jet. [Click here for additional details 2] The test particle is not drawn to the bottom stream because of a
velocity-dependent force that serves to repel a particle that is
moving in the same direction as the bottom stream. This
velocity-dependent gravitational effect is
gravitomagnetism.[18]:245–253
Matter in motion through a gravitomagnetic field is hence subject to
so-called frame-dragging effects analogous to electromagnetic
induction. It has been proposed that such gravitomagnetic forces
underlie the generation of the relativistic jets (Fig. 5‑8)
ejected by some rotating supermassive black holes.[54][55]
Return to Introduction
m a displaystyle m_ a ) is the mass which acts as the source of a gravitational field; (2) passive mass ( m p displaystyle m_ p ) is the mass which reacts to a gravitational field; (3) inertial mass ( m i displaystyle m_ i ) is the mass which reacts to acceleration.[58] m p displaystyle m_ p is the same as what we have earlier termed gravitational mass ( m g displaystyle m_ g ) in our discussion of the equivalence principle in the Basic propositions section. In Newtonian theory, The third law of action and reaction dictates that m a displaystyle m_ a and m p displaystyle m_ p must be the same. On the other hand, whether m p displaystyle m_ p and m i displaystyle m_ i are equal is an empirical result. In general relativity, The equality of m p displaystyle m_ p and m i displaystyle m_ i is dictated by the equivalence principle. There is no "action and reaction" principle dictating any necessary relationship between m a displaystyle m_ a and m p displaystyle m_ p .[58] Return to Introduction
•
Figure 5-9. (A)
The classic experiment to measure the strength of a gravitational source (i.e. its active mass) was first conducted in 1797 by Henry Cavendish (Fig. 5‑9a). Two small but dense balls are suspended on a fine wire, making a torsion balance. Bringing two large test masses close to the balls introduces a detectable torque. Given the dimensions of the apparatus and the measurable spring constant of the torsion wire, the gravitational constant G can be determined. To study pressure effects by compressing the test masses is hopeless, because attainable laboratory pressures are insignificant in comparison with the mass-energy of a metal ball. However, the repulsive electromagnetic pressures resulting from protons being tightly squeezed inside atomic nuclei are typically on the order of 1028 atm ≈ 1033 Pa ≈ 1033 kg·s−2m−1. This amounts to about 1% of the nuclear mass density of approximately 1018kg/m3 (after factoring in c2 ≈ 9×1016m2s−2).[59] Figure 5-10.
If pressure does not act as a gravitational source, then the ratio m a / m p displaystyle m_ a /m_ p should be lower for nuclei with higher atomic number Z, in which the
electrostatic pressures are higher. L. B. Kreuzer (1968) did a
Figure 5-11.
The existence of gravitomagnetism was proven by
( M , g ) displaystyle (M,g) . This means the smooth
g displaystyle g has signature ( 3 , 1 ) displaystyle (3,1) . The metric determines the geometry of spacetime, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold, coordinate charts are used to represent observers in reference frames. Usually, Cartesian coordinates ( x , y , z , t ) displaystyle (x,y,z,t) are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light c displaystyle c is equal to 1.[71] A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event p displaystyle p . Another reference frame may be identified by a second coordinate chart about p displaystyle p . Two observers (one in each reference frame) may describe the same event p displaystyle p but obtain different descriptions.[71] Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing p displaystyle p (representing an observer) and another containing q displaystyle q (representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.[71] For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event p displaystyle p ). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples ( x , y , z , t ) displaystyle (x,y,z,t) (as they are using different coordinate systems). Although their
kinematic descriptions will differ, dynamical (physical) laws, such as
momentum conservation and the first law of thermodynamics, will still
hold. In fact, relativity theory requires more than this in the sense
that it stipulates these (and all other physical) laws must take the
same form in all coordinate systems. This introduces tensors into
relativity, by which all physical quantities are represented.
Properties of n+m-dimensional spacetimes There are two kinds of dimensions, spatial (bidirectional) and
temporal (unidirectional[citation needed]). Let the number of spatial
dimensions be N and the number of temporal dimensions be T. That N = 3
and T = 1, setting aside the compactified dimensions invoked by string
theory and undetectable to date, can be explained by appealing to the
physical consequences of letting N differ from 3 and T differ from 1.
The argument is often of an anthropic character and possibly the first
of its kind, albeit before the complete concept came into vogue.
5 + 2 k displaystyle 5+2k spatial dimensions, where k is a whole number, then wave impulses
become distorted. In 1922,
In classical mechanics, time is separate from space. In special relativity, time and space are fused together into a single 4-dimensional "manifold" called spacetime. The technical term "manifold" and the great speed of light imply that at ordinary speeds, there is little that humans might observe which is noticeably different from what they would observe if the world followed the geometry of "common sense." Things that happen in spacetime are called "events". Events are idealized, four-dimensional points. There is no such thing as an event in motion. The path of a particle in spacetime traces out a succession of events, which is called the particle's "world line". In special relativity, to "observe" or "measure" an event means to ascertain its position and time against a hypothetical infinite latticework of synchronized clocks. To "observe" an event is not the same as to "see" an event. ^
To mid-1800s scientists, the wave nature of light implied a medium
that waved. Much research was directed to elucidate the properties of
this hypothetical medium, called the "luminiferous aether".
Experiments provided contradictory results. For example, stellar
aberration implied no coupling between matter and the aether, while
the
Return to Introduction
c displaystyle c so that the space and time coordinates have the same units (meters). ^Reference frames (click here to return to main) To simplify analyses of two reference frames in relative motion, Galilean (i.e. conventional 3-space) diagrams of the frames may be set in a standard configuration with aligned axes whose origins coincide when t = 0. A spacetime diagram in standard configuration is typically drawn with only a single space and a single time coordinate. The "unprimed frame" will have orthogonal x and ct axes. The axes of the "primed frame" will share a common origin with the unprimed axes, but its x' and ct' axes will be inclined by equal and opposite angles from the x and ct axes. Although the axes of the unprimed frame are orthogonal and the axes of the primed frame are inclined, the frames are actually equivalent. The asymmetry is due to unavoidable mapping distortions, and should be considered no stranger than the mapping distortions that occur, say, when mapping a spherical Earth onto a flat map. ^
On a spacetime diagram, two 45° diagonal lines crossing the origin represent light signals to and from the origin. In a diagram with an extra space direction, the diagonal lines form a "light cone". The light cone divides spacetime into a "timelike future" (separated from the origin by more time than space), a "timelike past", and an "elsewhere" region (separated from the origin by a "spacelike" interval with more space than time). Events in the future and past light cones are causally related to the origin. Events in the elsewhere region do not have a causal relationship with the origin. ^
If two events are timelike separated (causally related), then their before-after ordering is fixed for all observers. If two events are spacelike separated (non-causally related), then different observers with different relative motions may have reverse judgments on which event occurred before the other. Simultaneous events are necessarily spacelike separated. The spacetime interval between two simultaneous events gives the "proper distance". The spacetime interval measured along a world line gives the "proper time". ^Invariant hyperbola (click here to return to main) In a plane, the set of points equidistant from the origin form a circle. In a spacetime diagram, a set of points at a fixed spacetime interval from the origin forms an invariant hyperbola. The loci of points at constant spacelike and timelike intervals from the origin form timelike and spacelike invariant hyperbolae. ^
If frame S' is in relative motion to frame S, its ct' axis is tilted with respect to ct. Because of this tilt, one light-second on the ct' axis maps to greater than one light-second on the ct axis. Likewise, one light-second on the ct axis maps to greater than one light-second on the ct' axis. Each observer measures the other's clocks as running slow. The world sheet of a rod one light-second in length aligned parallel to the x' axis projects to less than one light-second on the x axis. Likewise, the world sheet of a rod one light-second in length aligned parallel to the x axis projects to less than one light-second on the x' axis. Each observer measures the other's rulers as being foreshortened. ^Mutual time dilation and the twin paradox (click here to return to main) ^Mutual time dilation (click here to return to main) To beginners, mutual time dilation seems self-contradictory because two observers in relative motion will each measure the other's clock as running more slowly. Careful consideration of how time measurements are performed reveals that there is no inherent necessity for the two observers' measurements to be reciprocally "consistent." In order to measure the rate of ticking of one of B's clocks, observer A must use two of his own clocks to record the time where B's clock made a first tick, and the time where B's clock made a second tick, so that a grand total of three clocks are involved in the measurement. Conversely, observer B uses three clocks to measure the rate of ticking of one of A's clocks. A and B are not doing the same measurement with the same clocks. ^
In the twin paradox, one twin A makes a journey into space in a high-speed rocket, returning home to find that the twin B who remained on Earth has aged more. The twin paradox is not a paradox because the twins' paths through spacetime are not equivalent. Throughout both the outbound and the inbound legs of the traveling twin's journey, A measures B's clocks as running slower than A's own. But during the turnaround, a shift takes place in the events of A's world line that B considers to be simultaneous with his own. ^Gravitation (click here to return to main) In the absence of gravity, spacetime is flat, is uniform throughout,
and serves as nothing more than a static background for the events
that take place in it.
Return to Introduction Basic mathematics of spacetime summary[edit] ^Galilean transformations (click here to return to main) A basic goal is to be able to compare measurements made by observers in relative motion. Transformation between Galilean frames is linear. Given that two coordinate systems are in standard configuration, the coordinate transformation in the x-axis is simply x ′ = x − v t displaystyle x'=x-vt Velocities are simply additive. If frame S' is moving at velocity v with respect to frame S, and within frame S', observer O' measures an object moving with velocity u', then u ′ = u − v displaystyle u'=u-v or u = u ′ + v displaystyle u=u'+v ^Relativistic composition of velocities (click here to return to main) The relativistic composition of velocities is more complex than the Galilean composition of velocities: u = v + u ′ 1 + ( v u ′ / c 2 ) . displaystyle u= v+u' over 1+(vu'/c^ 2 ) . In the low speed limit, the overall result is indistinguishable from the Galilean formula. The sum of two velocities cannot be greater than the speed of light. ^
The Lorentz factor, gamma γ , displaystyle gamma , appears very frequently in relativity. Given β = v / c , displaystyle beta =v/c, γ = 1 1 − v 2 / c 2 = 1 1 − β 2 displaystyle gamma = frac 1 sqrt 1-v^ 2 /c^ 2 = frac 1 sqrt 1-beta ^ 2 γ displaystyle gamma is the time dilation factor, while 1 / γ displaystyle 1/gamma is the length contraction factor. The Lorentz factor is undefined for v ≥ c . displaystyle vgeq c. ^Lorentz transformations (click here to return to main) The Lorentz transformations combine expressions for time dilation, length contraction, and relativity of simultaneity into a unified set of expressions for mapping measurements between two inertial reference frames. Given two coordinate systems in standard configuration, the transformation equations for the t displaystyle t and x displaystyle x axes are: t ′ = γ ( t − v x c 2 ) x ′ = γ ( x − v t ) displaystyle begin aligned t'&=gamma left(t- frac vx c^ 2 right)\x'&=gamma left(x-vtright)end aligned There have been many alternative derivations of the Lorentz
transformations since Einstein's original work in 1905.
The Lorentz transformations have a mathematical property called
linearity. Because of this: (i)
^
The formulas for classical
f = 1 − β 1 + β f 0 . displaystyle f= sqrt frac 1-beta 1+beta ,f_ 0 . Transverse Doppler shift is a relativistic effect that has no classical analog. Although there are subtleties involved, the basic scenarios can be analyzed by simple time dilation arguments. ^Energy and momentum (click here to return to main) In relativistic mechanics, the momentum vector is extended to four dimensions. Added to the momentum vector is a time component that allows the spacetime momentum vector to transform like the spacetime position vector. The relativistic energy-momentum vector has terms for energy and for spatial momentum. Using an uppercase P to represent the four-momentum and a lowercase p to denote the spatial momentum, the four-momentum may be written as P ≡ ( E / c , p → ) = ( E / c , p x , p y , p z ) displaystyle Pequiv (E/c, vec p )=(E/c,p_ x ,p_ y ,p_ z ) Consideration of the interrelationships between the various components of the relativistic momentum vector led Einstein to his famous E = m c 2 displaystyle E=mc^ 2 equation as well as to his concept of relativistic mass. ^Conservation laws (click here to return to main) The conservation laws arise from fundamental symmetries of nature. Classical conservation of mass does not hold true in relativity. Since mass and energy are interconvertible, conservation of mass is replaced by conservation of mass-energy. For analysis of energy and momentum problems involving interacting particles, the most convenient frame is usually the "center-of-momentum" frame. Newtonian momenta, calculated as p = m v , displaystyle p=mv, fail to behave properly under Lorentzian transformation. The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass. Return to Introduction
Beyond the basics summary[edit]
^
The formulas to perform
β displaystyle beta is the analog of slope. The rapidity, φ, is defined by β ≡ tanh ϕ ≡ v c displaystyle beta equiv tanh phi equiv frac v c Many expressions in special relativity take on a considerably simpler form when expressed in terms of rapidity. For example, the relativistic composition of velocities becomes simply ϕ = ϕ 1 + ϕ 2 . displaystyle phi =phi _ 1 +phi _ 2 . The Lorentz boost in the x direction becomes a hyperbolic rotation: ( c t ′ x ′ ) = ( cosh ϕ − sinh ϕ − sinh ϕ cosh ϕ ) ( c t x ) displaystyle begin pmatrix ct'\x'end pmatrix = begin pmatrix cosh phi &-sinh phi \-sinh phi &cosh phi end pmatrix begin pmatrix ct\xend pmatrix . ^4‑vectors (click here to return to main)
^Acceleration (click here to return to main) It is a common misconception that special relativity is unable to handle accelerating objects or accelerating reference frames. Special relativity handles such situations quite well. It is only when gravitation is significant that general relativity is required. The Dewan–Beran–Bell spaceship paradox is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues. The issues become almost trivial when analyzed with the aid of spacetime diagrams. Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as event horizons. Return to Introduction Introduction to curved spacetime summary[edit] ^Basic propositions (click here to return to main)
^Curvature of time (click here to return to main)
^Curvature of space (click here to return to main) Curvature of time completely accounts for all Newtonian gravitational effects. There are curvature terms for the spatial components of the invariant interval as well, but the effects on planetary orbits and the like are tiny. This is because the speeds of planets and satellites in their orbits are very much slower than the speed of light. Nevertheless, Urbain Le Verrier, in 1859, was able to demonstrate discrepancies in the orbit of Mercury from that predicted by Newton's laws. Einstein showed that this discrepancy, the anomalous precession of Mercury, is explained by the spatial terms in the curvature of spacetime. For light, the spatial terms in the invariant interval are comparable in magnitude to the temporal term, so the effects of the curvature of space are comparable to the effects of the curvature of time. The famous 1919 Eddington eclipse expedition showed that the bending of light around the Sun includes a component explained by the curvature of space. ^Sources of spacetime curvature (click here to return to main) In Newton's theory of gravitation, the only source of gravitational
force is mass. In contrast, general relativity identifies several
sources of spacetime curvature in addition to mass: Mass-energy
density, momentum density, pressure, and shear stress.
Return to Introduction See also[edit]
Basic introduction to the mathematics of curved spacetime Global spacetime structure Metric space Philosophy of space and time Notes[edit] ^ luminiferous from the Latin lumen, light, + ferens, carrying; aether
from the Greek αἰθήρ (aithēr), pure air, clear sky
^ By stating that simultaneity is a matter of convention, Poincaré
meant that to talk about time at all, one must have synchronized
clocks, and the synchronization of clocks must be established by a
specified, operational procedure (convention). This stance represented
a fundamental philosophical break from Newton, who conceived of an
absolute, true time that was independent of the workings of the
inaccurate clocks of his day. This stance also represented a direct
attack against the influential philosopher Henri Bergson, who argued
that time, simultaneity, and duration were matters of intuitive
understanding. Galison (2003), op cit.
^ The operational procedure adopted by Poincaré was essentially
identical to what is known as Einstein synchronization, even though a
variant of it was already a widely used procedure by telegraphers in
the middle 19th century. Basically, to synchronize two clocks, one
flashes a light signal from one to the other, and adjusts for the time
that the flash takes to arrive. Galison (2003), op cit.
^ In the original version of this lecture, Minkowski continued to use
such obsolescent terms as the ether, but the posthumous publication in
1915 of this lecture in the Annals of
Additional details[edit] ^ Different reporters viewing the scenarios presented in this figure
interpret the scenarios differently depending on their knowledge of
the situation. (i) A first reporter, at the center of mass of
particles 2 and 3 but unaware of the large mass 1, concludes that
a force of repulsion exists between the particles in scenario A
while a force of attraction exists between the particles in
scenario B. (ii) A second reporter, aware of the large
mass 1, smiles at the first reporter's naiveté. This second
reporter knows that in reality, the apparent forces between particles
2 and 3 really represent tidal effects resulting from their
differential attraction by mass 1. (iii) A third reporter,
trained in general relativity, knows that there are, in fact, no
forces at all acting between the three objects. Rather, all three
objects move along geodesics in spacetime.
^
References[edit] ^ Rynasiewicz, Robert. "Newton's Views on Space, Time, and Motion".
Stanford Encyclopedia of Philosophy. Metaphysics Research Lab,
Stanford University. Retrieved 24 March 2017.
^ Davis, Philip J. (2006). Mathematics & Common Sense: A Case of
Creative Tension. Wellesley, Massachusetts: A.K. Peters. p. 86.
ISBN 9781439864326.
^ a b c d e Collier, Peter (2014). A Most Incomprehensible Thing:
Notes Towards a Very Gentle Introduction to the Mathematics of
Relativity (2nd ed.). Incomprehensible Books.
ISBN 9780957389458.
^ Rowland, Todd. "Manifold". Wolfram Mathworld. Wolfram Research.
Retrieved 24 March 2017.
^ a b French, A.P. (1968).
Further reading[edit] Barrow, John D.; Tipler, Frank J. (1988). The Anthropic Cosmological
Principle. Oxford University Press. ISBN 978-0-19-282147-8.
LCCN 87028148.
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