The relativity principle states that the laws of physics are the same in all rigid inertial frames, so observers in different inertial frames should be able to compare measurements after suitable coordinate transformations.
Figure C.1 shows the inertial coordinate system and the “primed” system moving with a constant velocity along their common -axis. The planes and always coincide with the planes and , and clocks in both frames were synchronized by setting at the instant when .
An event is an observable confined to one point in space and time, such as a camera flash firing. The event coordinates in and are and , respectively.
According to “everyday” Galilean relativity, the event coordinates in are related to those in by the Galilean transform
However, everyday experience with slowly moving objects cannot be extrapolated to speeds approaching the vacuum speed of light . Everyday experience suggests that in all inertial frames, so Isaac Newton believed in “absolute, true, and mathematical time, in and of itself and of its own nature, without reference to anything external.” If and , parallel velocities simply add. The speed of a photon emitted in the -direction by a flash at rest in the system will be seen by an observer in the frame as
Thus Galilean relativity is inconsistent with both observation and Maxwell’s equations, which correctly predict that the speed of light in a vacuum is the same for all observers in all inertial frames (), regardless of their relative velocities . Apparently time is not absolute, and identical clocks in the and frames do not run at the same rates: .
The Lorentz transform is the only coordinate transform consistent with both relativity and the existence of some still-unspecified invariant speed (for example, the vacuum speed of light, or even ). The assumption that there exists some invariant speed is actually weaker than assuming because it turns out that reduces the Lorentz transform to the Galilean transform.
space is homogeneous, and
space is isotropic.
These assumptions say that the laws of physics are invariant under translation and rotation of the coordinate frames.
Homogeneity implies that any transformation from one inertial frame to another must be linear; that is, where and are constants. Adding nonlinear terms (e.g., ) would cause the transform itself to vary under coordinate translations, so in a homogeneous space. Because the coordinate frames were chosen such that when , linearity also requires that , leaving , where is a still-unspecified scale factor.
Isotropy implies that the observers in both frames agree on their relative speed because a 180 coordinate rotation exchanges the roles of the two frames. That rotation should have no effect if space is isotropic, so implies and only is consistent with isotropy. The negative solution can be rejected because it implies even when . These arguments can also be applied to and , so the Lorentz transform for the and coordinates is
in agreement with the Galilean transform (Equation C.1).
For the -coordinate, linearity requires
where and are still-unspecified constant scale factors. In Galilean relativity, .
Invoking isotropy and reversing the directions of and gives
and reversing the roles of the two frames gives
We now make use of the assumption that there is some speed which is the same in all inertial frames. Maxwell’s equations and experiment both show that is the speed of light in a vacuum, but for this argument, the invariant speed might be any speed, even , in which limit the Lorentz transform turns out to be identical to the Galilean transform. Then implies and Equation C.5 implies
of these two equations can be solved for the Lorentz factor:
where the dimensionless velocity is defined as
Again, the negative solution to Equation C.10 can be rejected as unphysical. Notice that for all possible . The -coordinate Lorentz transform becomes
Eliminating or from this pair of equations yields the Lorentz time transform
In summary, the Lorentz coordinate transform of special relativity is
Note that in the limit (), the Lorentz transform reduces to the Galilean transform (Equation C.1), as it must to agree with “everyday” observations involving small velocities. Equations C.14 and C.15 also show that the Lorentz transform reduces to the Galilean transform for any finite in the limit , so the assumption that there exists some invariant velocity is not restrictive.
If and are the coordinate differences between two events, the differential Lorentz transform is
The differences can be finite because the Lorentz transform is linear. This makes the differential Lorentz transform easy to apply to physical problems such as determining the lengths of rulers.
The phenomenon of relativistic time dilation follows from the differential time transform . If successive ticks of a clock at rest in the primed frame (so ) are separated by in time, they will be separated by in the unprimed frame. Likewise a clock at rest in the unprimed frame appears to run slow by the same factor when observed in the primed frame.
Relativistic length contraction can also be derived from Equation C.16. Suppose the ends of a ruler of unit length in the primed frame () emit two flashes of light at the same time in the unprimed frame (). The time between the flashes in the primed frame can be calculated from to be . The ruler length in the unprimed frame is the distance between the flashes. It is shorter by the factor
A ruler at rest in the unprimed frame also appears to be shorter by the factor when observed in the primed frame.
Relativistic velocities do not add linearly. Let the velocity of a particle be in the unprimed frame and in the primed frame:
The differential Lorentz transforms and yield
and, by symmetry,
For the velocity components perpendicular to ,
The ratio of a moving object’s relativistic mass to its rest mass follows from a thought experiment in Rindler . Imagine two identical electrons, one at rest in the unprimed frame and the other at rest in the primed frame moving with velocity . Let one electron be slightly displaced from the other along the -axis. At the moment the electrons pass by each other, their Coulomb repulsion will accelerate them in the -directions. Because , both electrons must experience the same displacement, but time dilation will make the moving electron take a factor longer to do so. Invoking momentum conservation, both observers conclude that the mass of the electron at rest in the moving frame is a factor larger:
Likewise, its total energy
has been multiplied by . Applying the chain rule for derivatives to the mass–energy transform,
shows that power is a relativistic invariant.