Lorentz Transformation

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Lorentz Transformation

The primed frame moves with velocity v in the x direction with respect to the fixed reference frame. The reference frames coincide at t=t'=0. The point x' is moving with the primed frame.

The reverse transformation is:

Much of the literature of relativity uses the symbols ? and ? as defined here to simplify the writing of relativistic relationships.

Length Contraction

The length of any object in a moving frame will appear foreshortened in the direction of motion, or contracted. The amount of contraction can be calculated from the Lorentz transformation. The length is maximum in the frame in which the object is at rest.

Time Dilation

A clock in a moving frame will be seen to be running slow, or "dilated" according to the Lorentz transformation. The time will always be shortest as measured in its rest frame. The time measured in the frame in which the clock is at rest is called the "proper time".

For small velocities at which the relativity factor is very close to 1, then the time dilation can be expanded in a binomial expansion to get the approximate expression:

Relativistic Mass

The increase in effective mass with speed is given by the expression

It follows from the Lorentz transformation when collisions are described from a fixed and moving reference frame, where it arises as a result of conservation of momentum

The increase in relativistic effective mass makes the speed of light c the speed limit of the universe. This increased effective mass is evident in cyclotrons and other accelerators where the speed approaches c. Exploring the calculation above will show that you have to reach 14% of the speed of light, or about 42 million m/s before you change the mass by 1%.

Relativistic Mass Example

At the electron accelerator in Cambridge, Mass., the final acceleration stage has the following characteristics:

	Feed electrons	Electrons out
Velocity	0.99986 c	0.999999996 c
Mass	60 m0	11,180 m0
Relative time for auto trip	2 hr	1 hr 59 min 59 sec

This increase in velocity requires a 186x increase in energy, yet only saves one second off a two hour journey.

Relativistic Momentum

The relativistic momentum is given by

which is the ordinary definition of momentum with the mass replaced by the relativistic mass.

Relativistic Energy

The famous Einstein relationship for energy

includes both the kinetic energy and rest mass energy for a particle. The kinetic energy of a high speed particle can be calculated from

The relativistic energy of a particle can also be expressed in terms of its momentum in the expression

The relativistic energy expression is the tool used to calculate binding energies of nuclei and the energy yields of nuclear fission and fusion.

Momentum of Photon

For a photon, the relativistic momentum expression

approaches zero over zero, so it can't be used directly to determine the momentum of a zero rest mass particle. But the general energy expression can be put in the form

and by setting rest mass equal to zero and applying the Planck relationship, we get the momentum expression:

Community Contributions - Articles by goIITians

Lorentz Transformation

Length Contraction

Time Dilation

Relativistic Mass

Relativistic Mass Example

Relativistic Momentum

Relativistic Energy

Momentum of Photon