Following is a list of the frequently occurring equations in the theory of special relativity.
To derive the equations of special relativity, one must start with two other
c0
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
c0
c0
c0
c0
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 x-direction, of a Cartesian coordinate system.
The following notations are used very often in special relativity:
\gamma=
| 1 | |
| \sqrt{1-\beta2 |
where
\beta=
| v | |
| c |
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 → ∞.
t'=\gammat
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.
\ell'=
| \ell | |
| \gamma |
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 . 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.
x'=\gamma\left(x-vt\right)
y'=y
z'=z
t'=\gamma\left(t-
| vx | |
| c2 |
\right)
| V' | |||||||||
|
| V' | |||||||||
|
| V' | |||||||||
|
See main article: metric tensor and four-vectors.
In what follows, bold sans serif is used for 4-vectors while normal bold roman is used for ordinary 3-vectors.
\boldsymbol{a
where
η
η=\begin{pmatrix}-1&0&0&0\ 0&1&0&0\ 0&0&1&0\ 0&0&0&1\end{pmatrix}
ds2=dx2+dy2+dz2-c2dt2=\begin{pmatrix}cdt&dx&dy&dz\end{pmatrix}\begin{pmatrix}-1&0&0&0\ 0&1&0&0\ 0&0&1&0\ 0&0&0&1\end{pmatrix}\begin{pmatrix}cdt\ dx\ dy\ dz\end{pmatrix}
In the above, ds2 is known as the spacetime interval. This inner product is invariant under the Lorentz transformation, that is,
η(\boldsymbol{a
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 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.
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:
x'=\gammax-\gamma\betact
y'=y
z'=z
ct'=\gammact-\gamma\betax
then in matrix form:
\begin{pmatrix}ct'\ x'\ y'\ z'\end{pmatrix}=\begin{pmatrix}\gamma&-\gamma\beta&0&0\ -\gamma\beta&\gamma&0&0\ 0&0&1&0\ 0&0&0&1\end{pmatrix}\begin{pmatrix}ct\ x\ y\ z\end{pmatrix}
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:
\boldsymbol{a
In the above,
\boldsymbol{a
\boldsymbol{a
\boldsymbol{a
Invariance and unification of physical quantities both arise from four-vectors.[1] 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.
| Property/effect | 3-vector | 4-vector | Invariant result | ||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Space-time events | 3-position: r = (x1, x2, x3) r ⋅ r\equivr2\equiv
+
+
| 4-position: X = (ct, x1, x2, x3) | \boldsymbol{X \begin{align}&\left(ct\right)2-\left(
+
+
\right)\\ &=\left(ct\right)2-r2\\ &=-\chi2=\left(c\tau\right)2\end{align}\ | τ = proper time χ = proper distance | |||||||||||||||||||||||||||||||||||||||||||
| Momentum-energy invariance | p=\gammamu 3-momentum: p = (p1, p2, p3) p ⋅ p\equivp2\equiv
+
+
\ | 4-momentum: P = (E/c, p1, p2, p3) \boldsymbol{P | \boldsymbol{P \begin{align}&\left(
\right)2-\left(
+
+
\right)\\ &=\left(
\right)2-p2\\ &=\left(mc\right)2\end{align}\ | which leads to: E2=\left(pc\right)2+\left(mc2\right)2 E = total energy | |||||||||||||||||||||||||||||||||||||||||||
| Velocity | 3-velocity: u = (u1, u2, u3) u=
| 4-velocity: U = (U0, U1, U2, U3) \boldsymbol{U | \boldsymbol{U | ||||||||||||||||||||||||||||||||||||||||||||
| Acceleration | 3-acceleration: a = (a1, a2, a3) a=
| 4-acceleration: A = (A0, A1, A2, A3) \boldsymbol{A | \boldsymbol{A | ||||||||||||||||||||||||||||||||||||||||||||
| Force | 3-force: f = (f1, f2, f3) f=
| 4-force: F = (F0, F1, F2, F3) \boldsymbol{F | \boldsymbol{F | ||||||||||||||||||||||||||||||||||||||||||||
See main article: Relativistic Doppler effect.
General doppler shift:
\nu'=\gamma\nu\left(1-\beta\cos\theta\right)
Doppler shift for emitter and observer moving right towards each other (or directly away):
\nu'=\nu
| \sqrt{1-\beta | |
Doppler shift for emitter and observer moving in a direction perpendicular to the line connecting them:
\nu'=\gamma\nu