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. 任何物体在运动参考系中的长度在运动方向上会显得缩短，即发生收缩。这种收缩的量可以通过洛伦兹变换计算。物体在静止参考系中的长度是最大的。

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". 在运动参考系中，钟会被人看到走得慢，或者说根据洛伦兹变换发生"时间膨胀"。时间总是其静止参考系中测量的最短。在钟静止的参考系中测量的时间称为"固有时"。

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"

where the effective mass is given by the energy divided by c². This suggests that this "m" is really about the energy, and that the real mass is the rest mass m₀.

The increase in relativistic "effective mass" is associated with 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 effective mass by 1%.

This language about "effective mass" is included here because it has long been used in elementary textbooks to describe the increase in energy with velocity, and it has a certain appeal for the description of relativistic effects such as the Cambridge example. But you won't find it in modern physics textbooks. One comment from a classic work on relativity is

"Ouch! The concept of 'relativistic mass' is subject to misunderstanding. That's why we don't use it. First, it applies the name mass - belonging to the magnitude of a 4- vector - to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of space-time itself." (Taylor and Wheeler in Spacetime Physics)

"Relativistic Mass" Example

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

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

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." 引入没有明确定义的运动物体的质量概念是不好的。最好只引入‘静止质量’m的概念。不如引入动量和能量的表达式来代替引入M。

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.