Proper Acceleration
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In
relativity theory The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
, proper acceleration is the physical
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
(i.e., measurable acceleration as by an
accelerometer An accelerometer is a tool that measures proper acceleration. Proper acceleration is the acceleration (the rate of change of velocity) of a body in its own instantaneous rest frame; this is different from coordinate acceleration, which is accele ...
) experienced by an object. It is thus acceleration relative to a
free-fall In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on i ...
, or
inertial In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
, observer who is momentarily at rest relative to the object being measured. Gravitation therefore does not cause proper acceleration, because the same gravity acts equally on the inertial observer. As a consequence, all inertial observers always have a proper acceleration of zero. Proper acceleration contrasts with coordinate acceleration, which is dependent on choice of
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
s and thus upon choice of observers (see three-acceleration in special relativity). In the standard inertial coordinates of special relativity, for unidirectional motion, proper acceleration is the rate of change of
proper velocity In relativity, proper velocity (also known as celerity) w of an object relative to an observer is the ratio between observer-measured displacement vector \textbf and proper time elapsed on the clocks of the traveling object: :\textbf = \frac ...
with respect to coordinate time. In an inertial frame in which the object is momentarily at rest, the proper acceleration 3-vector, combined with a zero time-component, yields the object's
four-acceleration In the theory of relativity, four-acceleration is a four-vector (vector in four-dimensional spacetime) that is analogous to classical acceleration (a three-dimensional vector, see three-acceleration in special relativity). Four-acceleration has ap ...
, which makes proper-acceleration's magnitude
Lorentz-invariant In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same ...
. Thus the concept is useful: (i) with accelerated coordinate systems, (ii) at relativistic speeds, and (iii) in curved spacetime. In an accelerating rocket after launch, or even in a rocket standing at the gantry, the proper acceleration is the acceleration felt by the occupants, and which is described as
g-force The gravitational force equivalent, or, more commonly, g-force, is a measurement of the type of force per unit mass – typically acceleration – that causes a perception of weight, with a g-force of 1 g (not gram in mass measure ...
(which is ''not'' a force but rather an acceleration; see that article for more discussion of proper acceleration) delivered by the vehicle only. The "acceleration of gravity" ("force of gravity") never contributes to proper acceleration in any circumstances, and thus the proper acceleration felt by observers standing on the ground is due to the mechanical force ''from the ground'', not due to the "force" or "acceleration" of gravity. If the ground is removed and the observer allowed to free-fall, the observer will experience coordinate acceleration, but no proper acceleration, and thus no g-force. Generally, objects in such a fall or generally any such ballistic path (also called inertial motion), including objects in orbit, experience no proper acceleration (neglecting small tidal accelerations for inertial paths in gravitational fields). This state is also known as "zero gravity" ("zero-g") or "free-fall," and it produces a sensation of
weightlessness Weightlessness is the complete or near-complete absence of the sensation of weight. It is also termed zero gravity, zero G-force, or zero-G. Weight is a measurement of the force on an object at rest in a relatively strong gravitational fi ...
. Proper acceleration reduces to coordinate acceleration in an inertial coordinate system in flat spacetime (i.e. in the absence of gravity), provided the magnitude of the object's proper-velocity (momentum per unit mass) is much less than the speed of light ''c''. Only in such situations is coordinate acceleration ''entirely'' felt as a g-force (i.e. a proper acceleration, also defined as one that produces measurable weight). In situations in which gravitation is absent but the chosen coordinate system is not inertial, but is accelerated with the observer (such as the accelerated reference frame of an accelerating rocket, or a frame fixed upon objects in a centrifuge), then g-forces and corresponding proper accelerations felt by observers in these coordinate systems are caused by the mechanical forces which resist their
weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
in such systems. This weight, in turn, is produced by
fictitious force A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame. It is related to Newton's second law of motion, which trea ...
s or "inertial forces" which appear in all such accelerated coordinate systems, in a manner somewhat like the weight produced by the "force of gravity" in systems where objects are fixed in space with regard to the gravitating body (as on the surface of the Earth). The total (mechanical) force that is calculated to induce the proper acceleration on a mass at rest in a coordinate system that has a proper acceleration, via Newton's law , is called the proper force. As seen above, the proper force is equal to the opposing reaction force that is measured as an object's "operational weight" (i.e. its weight as measured by a device like a spring scale, in vacuum, in the object's coordinate system). Thus, the proper force on an object is always equal and opposite to its measured weight.


Examples

When holding onto a carousel that turns at constant
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
an observer experiences a radially inward (
centripetal A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved trajectory, path. Its direction is always orthogonality, orthogonal to the motion of the body and towards the fixed po ...
) proper-acceleration due to the interaction between the handhold and the observer's hand. This cancels the radially outward ''geometric acceleration'' associated with their spinning coordinate frame. This outward acceleration (from the spinning frame's perspective) will become the coordinate acceleration when they let go, causing them to fly off along a zero proper-acceleration (
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
) path. Unaccelerated observers, of course, in their frame simply see their equal proper and coordinate accelerations vanish when they let go. Similarly, standing on a non-rotating planet (and on earth for practical purposes) observers experience an upward proper-acceleration due to the
normal force In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts, as in Figure 1. In this instance ''normal'' is used in the geometric sense and means perpendicular, as oppose ...
exerted by the earth on the bottom of their shoes. This cancels the downward geometric acceleration due to the choice of coordinate system (a so-called shell-frameEdwin F. Taylor and John Archibald Wheeler (2000) ''Exploring black holes'' (Addison Wesley Longman, NY) ). That downward acceleration becomes coordinate if they inadvertently step off a cliff into a zero proper-acceleration (geodesic or rain-frame) trajectory. Note that ''geometric accelerations'' (due to the connection term in the coordinate system's covariant derivative below) act on ''every gram of our being'', while proper-accelerations are usually caused by an external force. Introductory physics courses often treat gravity's downward (geometric) acceleration as due to a mass-proportional force. This, along with diligent avoidance of unaccelerated frames, allows them to treat proper and coordinate acceleration as the same thing. Even then if an object maintains a ''constant proper-acceleration'' from rest over an extended period in flat spacetime, observers in the rest frame will see the object's coordinate acceleration decrease as its coordinate velocity approaches lightspeed. The rate at which the object's proper-velocity goes up, nevertheless, remains constant. Thus the distinction between proper-acceleration and coordinate accelerationcf. C. W. Misner, K. S. Thorne and J. A. Wheeler (1973) ''Gravitation'' (W. H. Freeman, NY) , section 1.6 allows one to track the experience of accelerated travelers from various non-Newtonian perspectives. These perspectives include those of accelerated coordinate systems (like a carousel), of high speeds (where proper and coordinate times differ), and of curved spacetime (like that associated with gravity on Earth).


Classical applications

At low speeds in the inertial coordinate systems of
Newtonian physics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mec ...
, proper acceleration simply equals the coordinate acceleration a=d2x/dt2. As reviewed above, however, it differs from coordinate acceleration if one chooses (against Newton's advice) to describe the world from the perspective of an accelerated coordinate system like a motor vehicle accelerating from rest, or a stone being spun around in a slingshot. If one chooses to recognize that gravity is caused by the curvature of spacetime (see below), proper acceleration differs from coordinate acceleration in a
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
. For example, an object subjected to physical or proper acceleration ao will be seen by observers in a coordinate system undergoing constant acceleration aframe to have coordinate acceleration: \vec_\text = \vec_\text - \vec_\text. Thus if the object is accelerating with the frame, observers fixed to the frame will see no acceleration at all. Similarly, an object undergoing physical or proper acceleration ao will be seen by observers in a frame rotating with angular velocity to have coordinate acceleration: \vec_\text = \vec_\text - \vec\omega \times (\vec\omega \times \vec ) - 2 \vec\omega \times \vec_\text - \frac \times \vec. In the equation above, there are three geometric acceleration terms on the right-hand side. The first "centrifugal acceleration" term depends only on the radial position and not the velocity of our object, the second "Coriolis acceleration" term depends only on the object's velocity in the rotating frame but not its position, and the third "Euler acceleration" term depends only on position and the rate of change of the frame's angular velocity. In each of these cases, physical or proper acceleration differs from coordinate acceleration because the latter can be affected by your choice of coordinate system as well as by physical forces acting on the object. Those components of coordinate acceleration ''not'' caused by physical forces (like direct contact or electrostatic attraction) are often attributed (as in the Newtonian example above) to forces that: (i) act on every gram of the object, (ii) cause mass-independent accelerations, and (iii) don't exist from all points of view. Such geometric (or improper) forces include Coriolis forces,
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
forces,
g-forces The gravitational force equivalent, or, more commonly, g-force, is a measurement of the type of force per unit mass – typically acceleration – that causes a perception of weight, with a g-force of 1 g (not gram in mass measure ...
,
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
s and (as we see below)
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
forces as well.


Viewed from a flat spacetime slice

Proper-acceleration's relationships to coordinate acceleration in a specified slice of flat spacetime follow from Minkowski's flat-space metric equation . Here a single reference frame of yardsticks and synchronized clocks define map position x and map time ''t'' respectively, the traveling object's clocks define
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
''τ'', and the "d" preceding a coordinate means infinitesimal change. These relationships allow one to tackle various problems of "anyspeed engineering", albeit only from the vantage point of an observer whose extended map frame defines simultaneity.


Acceleration in (1+1)D

In the unidirectional case i.e. when the object's acceleration is parallel or antiparallel to its velocity in the spacetime slice of the observer, proper acceleration α and coordinate acceleration a are related through the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
by . Hence the change in proper-velocity w=dx/dτ is the integral of proper acceleration over map-time t i.e. for constant . At low speeds this reduces to the well-known relation between coordinate
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
and coordinate acceleration times map-time, i.e. Δ''v''=''a''Δ''t''. For constant unidirectional proper-acceleration, similar relationships exist between
rapidity In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with di ...
''η'' and elapsed proper time Δ''τ'', as well as between Lorentz factor ''γ'' and distance traveled Δ''x''. To be specific: \alpha=\frac=c \frac=c^2 \frac, where the various velocity parameters are related by \eta = \sinh^\left(\frac\right) = \tanh^\left(\frac\right) = \pm \cosh^\left(\gamma\right) . These equations describe some consequences of accelerated travel at high speed. For example, imagine a spaceship that can accelerate its passengers at "1 gee" (10 m/s2 or about 1.0 light year per year squared) halfway to their destination, and then decelerate them at "1 gee" for the remaining half so as to provide earth-like artificial gravity from point A to point B over the shortest possible time. For a map-distance of Δ''x''AB, the first equation above predicts a midpoint Lorentz factor (up from its unit rest value) of . Hence the round-trip time on traveler clocks will be , during which the time elapsed on map clocks will be . This imagined spaceship could offer round trips to
Proxima Centauri Proxima Centauri is a small, low-mass star located away from the Sun in the southern constellation of Centaurus. Its Latin name means the 'nearest tarof Centaurus'. It was discovered in 1915 by Robert Innes and is the nearest-kno ...
lasting about 7.1 traveler years (~12 years on Earth clocks), round trips to the
Milky Way The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye ...
's central
black hole A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
of about 40 years (~54,000 years elapsed on earth clocks), and round trips to
Andromeda Galaxy The Andromeda Galaxy (IPA: ), also known as Messier 31, M31, or NGC 224 and originally the Andromeda Nebula, is a barred spiral galaxy with the diameter of about approximately from Earth and the nearest large galaxy to the Milky Way. The gala ...
lasting around 57 years (over 5 million years on Earth clocks). Unfortunately, sustaining 1-gee acceleration for years is easier said than done, as illustrated by the maximum payload to launch mass ratios shown in the figure at right.


In curved spacetime

In the language of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the components of an object's acceleration four-vector ''A'' (whose magnitude is proper acceleration) are related to elements of the
four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
via a covariant derivative ''D'' with respect to proper time : A^\lambda := \frac = \frac + \Gamma^\lambda _U^\mu U^\nu Here ''U'' is the object's
four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
, and ''Γ'' represents the coordinate system's 64 connection coefficients or
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
. Note that the Greek subscripts take on four possible values, namely 0 for the time-axis and 1-3 for spatial coordinate axes, and that repeated indices are used to indicate
summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, mat ...
over all values of that index. Trajectories with zero proper acceleration are referred to as
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s. The left hand side of this set of four equations (one each for the time-like and three spacelike values of index λ) is the object's proper-acceleration 3-vector combined with a null time component as seen from the vantage point of a reference or book-keeper coordinate system in which the object is at rest. The first term on the right hand side lists the rate at which the time-like (energy/''mc'') and space-like (momentum/''m'') components of the object's four-velocity ''U'' change, per unit time ''τ'' on traveler clocks. Let's solve for that first term on the right since at low speeds its spacelike components represent the coordinate acceleration. More generally, when that first term goes to zero the object's coordinate acceleration goes to zero. This yields \frac =A^\lambda - \Gamma^\lambda _U^\mu U^\nu. Thus, as exemplified with the first two animations above, coordinate acceleration goes to zero whenever proper-acceleration is exactly canceled by the connection (or ''geometric acceleration'') term on the far right. ''Caution:'' This term may be a sum of as many as sixteen separate velocity and position dependent terms, since the repeated indices ''μ'' and ''ν'' are by convention summed over all pairs of their four allowed values.


Force and equivalence

The above equation also offers some perspective on forces and the
equivalence principle In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (suc ...
. Consider ''local'' book-keeper coordinates for the metric (e.g. a local Lorentz tetrad like that which
global positioning system The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of the global navigation satellite sy ...
s provide information on) to describe time in seconds, and space in distance units along perpendicular axes. If we multiply the above equation by the traveling object's rest mass m, and divide by Lorentz factor ''γ'' = d''t''/d''τ'', the spacelike components express the rate of momentum change for that object from the perspective of the coordinates used to describe the metric. This in turn can be broken down into parts due to proper and geometric components of acceleration and force. If we further multiply the time-like component by lightspeed ''c'', and define coordinate velocity as , we get an expression for rate of energy change as well: :\frac=\vec\cdot\frac (timelike) and \frac = \sum \vec + \sum \vec = m(\vec+\vec) (spacelike). Here ''a''''o'' is an acceleration due to proper forces and ''a''''g'' is, by default, a geometric acceleration that we see applied to the object because of our coordinate system choice. At low speeds these accelerations combine to generate a coordinate acceleration like , while for unidirectional motion ''at any speed'' ''a''''o'''s magnitude is that of proper acceleration ''α'' as in the section above where ''α'' = ''γ''3''a'' when ''a''''g'' is zero. In general expressing these accelerations and forces can be complicated. Nonetheless if we use this breakdown to describe the connection coefficient (Γ) term above in terms of geometric forces, then the motion of objects from the point of view of ''any coordinate system'' (at least at low speeds) can be seen as locally Newtonian. This is already common practice e.g. with centrifugal force and gravity. Thus the equivalence principle extends the local usefulness of Newton's laws to accelerated coordinate systems and beyond.


Surface dwellers on a planet

For low speed observers being held at fixed radius from the center of a spherical planet or star, coordinate acceleration ashell is approximately related to proper acceleration ao by: \vec_\text = \vec_\text - \sqrt \frac \hat where the planet or star's
Schwarzschild radius The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteris ...
. As our shell observer's radius approaches the Schwarzschild radius, the proper acceleration ''a''o needed to keep it from falling in becomes intolerable. On the other hand, for , an upward proper force of only is needed to prevent one from accelerating downward. At the Earth's surface this becomes: \vec_\text = \vec_o - g \hat where is the downward 9.8 m/s2 acceleration due to gravity, and \hat is a unit vector in the radially outward direction from the center of the gravitating body. Thus here an outward proper force of mg is needed to keep one from accelerating downward.


Four-vector derivations

The spacetime equations of this section allow one to address ''all deviations'' between proper and coordinate acceleration in a single calculation. For example, let's calculate the
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
:Hartle, James B. (2003). Gravity: an Introduction to Einstein's General Relativity. San Francisco: Addison-Wesley. . \left( \begin \left\ & \left\ & \left\ & \left\ \\ \left\ & \left\ & \left\ & \left\ \\ \left\ & \left\ & \left\ & \left\ \\ \left\ & \left\ & \left\ & \left\ \end \right) for the far-coordinate Schwarzschild metric , where ''r''s is the
Schwarzschild radius The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteris ...
2''GM''/''c''2. The resulting array of coefficients becomes: \left( \begin \left\ & \left\ & \ & \ \\ \left\ & \left\ & \ & \left\ \\ \ & \left\ & \left\ & \ \\ \ & \left\ & \ & \left\ \end \right). From this you can obtain the shell-frame proper acceleration by setting coordinate acceleration to zero and thus requiring that proper acceleration cancel the geometric acceleration of a stationary object i.e. A^\lambda = \Gamma^\lambda _U^\mu U^\nu = \. This does not solve the problem yet, since
Schwarzschild coordinates In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of ''nested round spheres''. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coord ...
in curved spacetime are book-keeper coordinates but not those of a local observer. The magnitude of the above proper acceleration 4-vector, namely \alpha = \sqrtGM/r^2, is however precisely what we want i.e. the upward frame-invariant proper acceleration needed to counteract the downward geometric acceleration felt by dwellers on the surface of a planet. A special case of the above Christoffel symbol set is the flat-space
spherical coordinate In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
set obtained by setting ''r''s or ''M'' above to zero: \left( \begin \left\ & \left\ & \ & \ \\ \left\ & \left\ & \ & \left\ \\ \ & \left\ & \left\ & \ \\ \ & \left\ & \ & \left\ \end \right). From this we can obtain, for example, the centri''petal'' proper acceleration needed to cancel the centri''fugal'' geometric acceleration of an object moving at constant angular velocity at the equator where . Forming the same 4-vector sum as above for the case of and zero yields nothing more than the classical acceleration for rotational motion given above, i.e. A^\lambda = \Gamma^\lambda _U^\mu U^\nu = \ so that . Coriolis effects also reside in these connection coefficients, and similarly arise from coordinate-frame geometry alone.


See also

*
Acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
: change in velocity *
Proper velocity In relativity, proper velocity (also known as celerity) w of an object relative to an observer is the ratio between observer-measured displacement vector \textbf and proper time elapsed on the clocks of the traveling object: :\textbf = \frac ...
: momentum per mass in special relativity; composed of the spacelike components of the 4-velocity *
Proper reference frame (flat spacetime) A proper reference frame in the theory of relativity is a particular form of accelerated reference frame, that is, a reference frame in which an accelerated observer can be considered as being at rest. It can describe phenomena in curved spacetime, ...
: accelerated reference frame in special relativity (Minkowski space) *
Fictitious force A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame. It is related to Newton's second law of motion, which trea ...
: one name for mass times ''geometric acceleration'' *
Four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
: making the connection between space and time explicit *
Kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
: for studying ways that position changes with time * Uniform acceleration: holding coordinate acceleration fixed


Footnotes


External links

* Excerpts from the first edition of ''Spacetime Physics'', and othe
resources posted by Edwin F. Taylor

James Hartle's gravity book page
including Mathematica programs to calculate Christoffel symbols. * Andrew Hamilton'

for working with local tetrads at U. Colorado, Boulder. {{DEFAULTSORT:Proper Acceleration Minkowski spacetime Acceleration