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.


For v = c, L = L0
Lorentz transformation
Application in muon decay experiment
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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 v = c, T = T0

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:

See also Gravitational time dilation

Time dilation experiments
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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.

For v = c, m =m0

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%.

Cambridge exampleProblems with variable mass conceptMass of photon?
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Relativistic Mass Example

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

Feed electronsElectrons out
Velocity0.99986 c0.999999996 c
Mass60 m011,180 m0
Relative time for auto trip2 hr1 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.

Problems with variable mass concept

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Problems with variable mass

Even though circumstances like that described at the Cambridge accelerator are conveniently described by assuming an increasing mass, that is not the only way to describe these experiments, and there are problems with the concept of variable relativistic mass. Einstein's point of view is described in the following quote:

"It is not good to introduce the concept of the mass of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the 'rest mass' m. Instead of introducing M it is better to mention the expression for the momentum and energy of a body in motion."

Upon being introduced to special relativity for the first time, it is easier to contemplate concepts like the speed of light as the speed limit of the universe by envisioning the mass as increasing to infinity at velocity c. However, when one has become familiar with the concepts of relativistic momentum and relativistic energy, there is no real need for the variable mass concept.

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