The Lorentz transformation:
The matrix tensor:
The space time interval:
Other useful energy-momentum relations:
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:
From these two postulates, all of special relativity follows.
However, to the observer on the ground, the situation is very different. Because 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 two-right 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 , and then plugging in for , one finds:
This is the formula for time dilation. In particular, the following notations are used very often in special relativity:
This makes the formula for time dilation:
There are a few things worth pointing out here before moving on. First, is one for two frames at rest, and gets progressively larger the closer to the speed of light the velocity between the two frames becomes. At the speed of light, is effectively infinite. Second is that in this example the time measured in the frame on the vehicle, , 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.
However, the observer on the ground, making the same measurement, comes to a different conclusion. This observer finds that time 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 . 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, dimensions perpendicular to the direction of motion are unaffected by length contraction.
If you now plug in this result into the Galilean transformation, substituting 's for 's, and assuming the velocity is wholly in the -direction, you get:
And going from the primed frame to the unprimed frame:
Going from the primed frame to the unprimed frame was accomplished by making 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 figure out how and transform, you have to plug the -to- transformation into its reverse:
Plugging in the value for :
Finally, dividing through by :
Or more commonly:
And the converse can again be gotten by changing the sign of , and exchanging the unprimed variables for their primed counter-parts, and vice-versa. These transformations together are the Lorentz transformation:
Now velocity is , so
Now, putting in and , you get the velocity addition - actually, the formulae below are subtraction, but addition is just flipping the various signs around - formula of special relativity:
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 / transformation. The and equations were both arrived at by dividing the appropriate space differential (e.g. or ) by the time differential.
And, finally, 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 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 can be used when transforming between the same two frames.
As with four vectors, there is a concept of the dot product, or the inner product. The form of this is:
is known as the metric tensor. In special relativity, the metric tensor is as follows:
In the above, is known as the spacetime interval. Another thing worth noting is that this inner product is invariant under the Lorentz transformation. To have the inner product be invariant means the following:
On a final note about the matrix and four-vector formulation of special relativity, the sign of the metric and the placement of the , , , and time-based terms can vary, with different people working in different standards. For instance, many times the time-based terms are placed first in the four-vectors, with the spatial terms following. Also, sometimes is replaced with , making the spacial 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 through out the computations performed.
In the above, is the proper time of the path through spacetime, called the world-line, followed by the object velocity the above represents. As stated above in the derivation of time dilation, the proper time is the time between two events in a frame of reference where they take place at the same location. With the example of time dilation in mind, one can use the formula for it, as follows:
The first three terms, excepting the factor of , is the velocity as seen by the observer in their own reference frame. The is determined by the velocity between the observer's reference frame and the object's frame, which is the frame in which its proper time is measured. This quantity is invariant under Lorentz transformation, so to check to see what an observer in a different reference frame sees, one simply multiplies the velocity four-vector by the Lorentz transformation matrix between the two reference frames.
The four-momentum of an object is relatively straightforward:
This definition is identical in form to the relation to the classical momentum and the classical velocity. The mass, is also its usual self. For the spatial portion of the four-momentum one gets:
In the above, the factor of comes from the definition of the four-velocity described above. This formulation also has an alternative way of being stated:
This formulation makes the new relation between the spatial velocity and the spatial momentum look practically identical. However, this can be misleading, as it is not appropriate in special relativity in all circumstances. For instance, kinetic energy and force in special relativity can not be written exactly like their classical analogues by only replacing the mass with the relativistic mass. Moreover, under Lorentz transformations, this relativistic mass is not invariant, while the regular mass is. That is why many people find it easier to just stick with the regular mass, and discard the relativistic mass.
As it turn out, this is directly related to the energy in special relativity as follows:
This is the reason that the momentum four-vector is sometimes called the momentum-energy four-vector. However, it is not clear that this definition corresponds to the definition of the energy of a free particle classically, which is just the particle's kinetic energy. However, if you use the binomial series expansion of for small velocities, you see the following:
The second term above is the classical kinetic energy. The first term is something new entirely, and reduces to Einstein's famous equation when the object is not moving. This term is called the rest mass. In addition to the classical term and the rest mass term, there are more terms that could be added to the expansion, but they all involve factors of at least , which, for the approximation that the velocity is much less than the speed of light, mean that they are negligible and not easily detectable.
Taking the above expansion into account, noting that an object with zero velocity has energy equal to , one can show that the kinetic energy in special relativity is:
As the four-momentum is Lorentz invariant, the dot product of the four-momentum with itself is invariant under Lorentz transformations, so the above relation is true for the four-momentum in any frame of reference. In fact, assuming that there are spatial momentum terms, you can produce another relation:
In the above, is the relativistic three velocity.
You can simplify the last term by noting that , you can do the following:
It should be noted that in the above, it is assumed that the mass is constant over time. Plugging this into the first result:
This is the relativistic four-force, which is invariant under Lorentz transformation.
Now, if is , where is the angle between 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 x-axis. There are two special cases of this equation. The first is the case where the velocity between the emitter and observer is along the x-axis. In that case , and , which gives:
This is the equation for doppler shift in the case where the velocity between the emitter and observer is along the x-axis. The second special case is that where the relative velocity is perpendicular to the x-axis, and thus , and , which gives:
This is actually completely analogous to time dilation, as frequency is one over time. So, doppler shift for emitters and observers moving perpendicular to the line connecting them is completely due to the effects of time dilation.