Following is a list of the frequently occurring equations in the theory of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
.
Postulates of Special Relativity
To derive the equations of special relativity, one must start with two other
#The laws of physics are invariant under transformations between inertial frames. In other words, the laws of physics will be the same whether you are testing them in a frame 'at rest', or a frame moving with a constant velocity relative to the 'rest' frame.
#The speed of light in a perfect classical vacuum (
) is measured to be the same by all observers in inertial frames and is, moreover, finite but nonzero. This speed acts as a supremum for the speed of local transmission of information in the universe.
In this context, "speed of light" really refers to the speed supremum of information transmission or of the movement of ordinary (nonnegative mass) matter, locally, as in a classical vacuum. Thus, a more accurate description would refer to
rather than the speed of light per se. However, light and other massless particles do theoretically travel at
under vacuum conditions and experiment has nonfalsified this notion with fairly high precision. Regardless of whether light itself does travel at
, though
does act as such a supremum, and that is the assumption which matters for Relativity.
From these two postulates, all of special relativity follows.
In the following, the
relative velocity
The relative velocity \vec_ (also \vec_ or \vec_) is the velocity of an object or observer B in the rest frame of another object or observer A.
Classical mechanics
In one dimension (non-relativistic)
We begin with relative motion in the classi ...
''v'' between two
inertial frame
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. ...
s is restricted fully to the ''x''-direction, of a
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
.
Kinematics
Lorentz transformation
The following notations are used very often in special relativity:
;
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 ...
:
where
and ''v'' is the relative velocity between two
inertial frame
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. ...
s.
For two frames at rest, γ = 1, and increases with relative velocity between the two inertial frames. As the relative velocity approaches the speed of light, γ → ∞.
;
Time dilation
In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
(different times ''t'' and ''t at the same position ''x'' in same inertial frame)
:
:
In this example the time measured in the frame on the vehicle, ''t'', is known as the
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 ...
. The proper time between two events - such as the event of light being emitted on the vehicle and the event of light being received on the vehicle - is the time between the two events in a frame where the events occur at the same location. So, above, the emission and reception of the light both took place in the vehicle's frame, making the time that an observer in the vehicle's frame would measure the proper time.
;
Length contraction
Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald ...
(different positions ''x'' and ''x at the same instant ''t'' in the same inertial frame)
:
:
This is the formula for length contraction. As there existed a proper time for time dilation, there exists a
proper length
Proper length or rest length is the length of an object in the object's rest frame.
The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on t ...
for length contraction, which in this case is '. The proper length of an object is the length of the object in the frame in which the object is at rest. Also, this contraction only affects the dimensions of the object which are parallel to the relative velocity between the object and observer. Thus, lengths perpendicular to the direction of motion are unaffected by length contraction.
;
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
:
:
:
:
:
;
Velocity addition
In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different fra ...
:
:
:
:
The metric and four-vectors
In what follows, bold sans serif is used for
4-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 ...
s while normal bold roman is used for ordinary 3-vectors.
;
Inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
(i.e. notion of
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
):
:
where
is known as the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
. In special relativity, the metric tensor is the
Minkowski metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
:
:
;
Space-time interval
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
:
In the above, ''ds''
2 is known as the spacetime interval. This inner product is invariant under the Lorentz transformation, that is,
:
The sign of the metric and the placement of the ''ct'', ''ct, ''cdt'', and ''cdt′'' time-based terms can vary depending on the author's choice. For instance, many times the time-based terms are placed first in the four-vectors, with the spatial terms following. Also, sometimes ''η'' is replaced with −''η'', making the spatial terms produce negative contributions to the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
or spacetime interval, while the time term makes a positive contribution. These differences can be used in any combination, so long as the choice of standards is followed completely throughout the computations performed.
Lorentz transforms
It is possible to express the above coordinate transformation via a matrix. To simplify things, it can be best to replace ''t'', ''t′'', ''dt'', and ''dt′'' with ''ct'', ''ct, ''cdt'', and ''cdt′'', which has the dimensions of distance. So:
:
:
:
:
then in matrix form:
:
The vectors in the above transformation equation are known as four-vectors, in this case they are specifically the position four-vectors. In general, in special relativity, four-vectors can be transformed from one reference frame to another as follows:
:
In the above,
and
are the four-vector and the transformed four-vector, respectively, and Λ is the transformation matrix, which, for a given transformation is the same for all four-vectors one might want to transform. So
can be a four-vector representing position, velocity, or momentum, and the same Λ can be used when transforming between the same two frames. The most general Lorentz transformation includes boosts and rotations; the components are complicated and the transformation requires
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
s.
4-vectors and frame-invariant results
Invariance and unification of physical quantities both arise from
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 ...
s.
[Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Manchester Physics Series, John Wiley & Sons, 2009, ] The inner product of a 4-vector with itself is equal to a scalar (by definition of the inner product), and since the 4-vectors are physical quantities their magnitudes correspond to physical quantities also.
Doppler shift
General doppler shift:
:
Doppler shift for emitter and observer moving right towards each other (or directly away):
:
Doppler shift for emitter and observer moving in a direction perpendicular to the line connecting them:
:
:
See also
*
Theory of relativity
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 ...
*
Special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
*
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 ...
*
List of physics formulae
In physics, there are equations in every field to relate physical quantities to each other and perform calculations. Entire handbooks of equations can only summarize most of the full subject, else are highly specialized within a certain field. Phys ...
*
Defining equation (physics)
In physics, defining equations are equations that define new quantities in terms of base quantities. This article uses the current SI system of units, not natural or characteristic units.
Description of units and physical quantities
Physical q ...
*
Defining equation (physical chemistry)
In physical chemistry, there are numerous quantities associated with chemical compounds and reactions; notably in terms of ''amounts'' of substance, ''activity'' or ''concentration'' of a substance, and the ''rate'' of reaction. This article uses ...
*
Constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approx ...
*
List of equations in classical mechanics
Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known. The sub ...
*
Table of thermodynamic equations
This article is a summary of common equations and quantities in thermodynamics (see thermodynamic equations for more elaboration).
Definitions
Many of the definitions below are also used in the thermodynamics of chemical reactions.
General ...
*
List of equations in wave theory
This article summarizes equations in the theory of waves.
Definitions
General fundamental quantities
A wave can be longitudinal where the oscillations are parallel (or antiparallel) to the propagation direction, or transverse where the oscill ...
*
List of equations in gravitation
This article summarizes equations in the theory of gravitation.
Definitions
Gravitational mass and inertia
A common misconception occurs between centre of mass and centre of gravity. They are defined in similar ways but are not exactly the s ...
*
List of electromagnetism equations
This article summarizes equations in the theory of electromagnetism.
Definitions
Here subscripts ''e'' and ''m'' are used to differ between electric and magnetic charges. The definitions for monopoles are of theoretical interest, although rea ...
*
List of photonics equations
This article summarizes equations used in optics, including geometric optics, physical optics, radiometry, diffraction, and interferometry.
Definitions
Geometric optics (luminal rays)
General fundamental quantities
Physical optics (EM lum ...
*
List of equations in nuclear and particle physics
This article summarizes equations in the theory of nuclear physics and particle physics.
Definitions
Equations
Nuclear structure
Nuclear decay
Nuclear scattering theory
The following apply for the nuclear reaction:
:''a'' + ''b'' ↠...
References
Sources
* ''Encyclopaedia of Physics (2nd Edition)'',
R.G. Lerner, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) 0-89573-752-3
* ''Dynamics and Relativity'', J.R. Forshaw, A.G. Smith, Wiley, 2009,
* ''Relativity DeMystified'', D. McMahon, Mc Graw Hill (USA), 2006,
* ''The Cambridge Handbook of Physics Formulas'', G. Woan, Cambridge University Press, 2010, .
* ''An Introduction to Mechanics'', D. Kleppner, R.J. Kolenkow, Cambridge University Press, 2010, {{ISBN, 978-0-521-19821-9
Special relativity
Equations of physics
Relativistic equations
Following is a list of the frequently occurring equations in the theory of special relativity.
Postulates of Special Relativity
To derive the equations of special relativity, one must start with two other
#The laws of physics are invariant ...