List of 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 postulates:
 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 vacuum is measured to be the same by all observers in inertial frames.
From these two postulates, all of special relativity follows.
In the following, the relative velocity v between two inertial frames is restricted fully to the xdirection, of a Cartesian coordinate system.
Kinematics
Lorentz transformation
The following notations are used very often in special relativity:
where β = and v is the relative velocity between two inertial frames.
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 (different times t and t' at the same position x in same inertial frame)
Derivation of time dilation Applying the above postulates, consider the inside of any vehicle (usually exemplified by a train) moving with a velocity v with respect to someone standing on the ground as the vehicle passes. Inside, a light is shone upwards to a mirror on the ceiling, where the light reflects back down. If the height of the mirror is h, and the speed of light c, then the time it takes for the light to go up and come back down is:
However, to the observer on the ground, the situation is very different. Since the train is moving by the observer on the ground, the light beam appears to move diagonally instead of straight up and down. To visualize this, picture the light being emitted at one point, then having the vehicle move until the light hits the mirror at the top of the vehicle, and then having the train move still more until the light beam returns to the bottom of the vehicle. The light beam will have appeared to have moved diagonally upward with the train, and then diagonally downward. This path will help form tworight sided triangles, with the height as one of the sides, and the two straight parts of the path being the respective hypotenuses:
Rearranging to get :
Taking out a factor of c, and then plugging in for t, one finds:
This is the formula for time dilation:
In this example the time measured in the frame on the vehicle, t, is known as the proper time. 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 (different positions x and x' at the same instant t in the same inertial frame)
Derivation of length contraction Consider a long train, moving with velocity v with respect to the ground, and one observer on the train and one on the ground, standing next to a post. The observer on the train sees the front of the train pass the post, and then, some time t′ later, sees the end of the train pass the same post. He then calculates the train's length as follows:
However, the observer on the ground, making the same measurement, comes to a different conclusion. This observer finds that time t passed between the front of the train passing the post, and the back of the train passing the post. Because the two events  the passing of each end of the train by the post  occurred in the same place in the ground observer's frame, the time this observer measured is the proper time. So:
This is the formula for length contraction. As there existed a proper time for time dilation, there exists a proper length for length contraction, which in this case is ℓ. 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.
Derivation of Lorentz transformation using time dilation and length contraction Now substituting the length contraction result into the Galilean transformation (i.e. x = ℓ), we have:
that is:
and going from the primed frame to the unprimed frame:
Going from the primed frame to the unprimed frame was accomplished by making v in the first equation negative, and then exchanging primed variables for unprimed ones, and vice versa. Also, as length contraction does not affect the perpendicular dimensions of an object, the following remain the same as in the Galilean transformation:
Finally, to determine how t and t′ transform, substituting the x↔x′ transformation into its inverse:
Plugging in the value for γ:
Finally, dividing through by γv:
Or more commonly:
And the converse can again be gotten by changing the sign of v, and exchanging the unprimed variables for their primed variables, and vice versa. These transformations together are the Lorentz transformation:
 Velocity addition
Derivation of velocity addition The Lorentz transformations also apply to differentials, so:
The velocity is dx/dt, so
Now substituting:
gives the velocity addition (actually below is subtraction, addition is just reversing the signs of V_{x}, V_{y}, and V_{z} around):
Also, the velocities in the directions perpendicular to the frame changes are affected, as shown above. This is due to time dilation, as encapsulated in the dt/dt′ transformation. The V′_{y} and V′_{z} equations were both derived by dividing the appropriate space differential (e.g. dy′ or dz′) by the time differential.
The metric and fourvectors
In what follows, bold sans serif is used for 4vectors while normal bold roman is used for ordinary 3vectors.
 Inner product (i.e. notion of length)
where is known as the metric tensor. In special relativity, the metric tensor is the Minkowski metric:
 Spacetime interval
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′ timebased terms can vary depending on the author's choice. For instance, many times the timebased terms are placed first in the fourvectors, with the spatial terms following. Also, sometimes η is replaced with −η, making the spatial terms produce negative contributions to the dot product 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 fourvectors, in this case they are specifically the position fourvectors. In general, in special relativity, fourvectors can be transformed from one reference frame to another as follows:
In the above, and are the fourvector and the transformed fourvector, respectively, and Λ is the transformation matrix, which, for a given transformation is the same for all fourvectors one might want to transform. So can be a fourvector 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 spinors.
4vectors and frameinvariant results
Invariance and unification of physical quantities both arise from fourvectors.[1] The inner product of a 4vector with itself is equal to a scalar (by definition of the inner product), and since the 4vectors are physical quantities their magnitudes correspond to physical quantities also.
Property/effect  3vector  4vector  Invariant result 

Spacetime events  3position: r = (x_{1}, x_{2}, x_{3}) 
4position: X = (ct, x_{1}, x_{2}, x_{3}) 

Momentumenergy invariance 
3momentum: p = (p_{1}, p_{2}, p_{3}) 
4momentum: P = (E/c, p_{1}, p_{2}, p_{3}) 
which leads to: E = total energy 
Velocity  3velocity: u = (u_{1}, u_{2}, u_{3}) 
4velocity: U = (U_{0}, U_{1}, U_{2}, U_{3}) 

Acceleration  3acceleration: a = (a_{1}, a_{2}, a_{3}) 
4acceleration: A = (A_{0}, A_{1}, A_{2}, A_{3}) 

Force  3force: f = (f_{1}, f_{2}, f_{3}) 
4force: F = (F_{0}, F_{1}, F_{2}, F_{3}) 

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:
Derivation of the relativistic Doppler shift If an object emits a beam of light or radiation, the frequency, wavelength, and energy of that light or radiation will look different to a moving observer than to one at rest with respect to the emitter. If one assumes that the observer is moving with respect to the emitter along the xaxis, then the standard Lorentz transformation of the fourmomentum, which includes energy, becomes:
Now, if
where θ is the angle between p_{x} and , and plugging in the formulas for frequency's relation to momentum and energy:
This is the formula for the relativistic doppler shift where the difference in velocity between the emitter and observer is not on the xaxis. There are two special cases of this equation. The first is the case where the velocity between the emitter and observer is along the xaxis. In that case θ = 0, and cos θ = 1, which gives:
This is the equation for doppler shift in the case where the velocity between the emitter and observer is along the xaxis. The second special case is that where the relative velocity is perpendicular to the xaxis, and thus θ = π/2, and cos θ = 0, which gives:
This is actually completely analogous to time dilation, as frequency is the reciprocal of time. So, doppler shift for emitters and observers moving perpendicular to the line connecting them is completely due to the effects of time dilation.
See also
 Theory of relativity
 Special relativity
 General relativity
 List of physics formulae
 Defining equation (physics)
 Defining equation (physical chemistry)
 Constitutive equation
 List of equations in classical mechanics
 Table of thermodynamic equations
 List of equations in wave theory
 List of equations in gravitation
 List of electromagnetism equations
 List of photonics equations
 List of equations in quantum mechanics
 List of equations in nuclear and particle physics
References
 Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Manchester Physics Series, John Wiley & Sons, 2009, ISBN 9780470014608
Sources
 Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3527269541, (VHC Inc.) 0895737523
 Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 9780470014608
 Relativity DeMystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0071455450
 The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 9780521575072.
 An Introduction to Mechanics, D. Kleppner, R.J. Kolenkow, Cambridge University Press, 2010, ISBN 9780521198219