愛因斯坦相對論

狹義相對論

圖片參考：http://upload.wikimedia.org/math/2/e/b/2eb931bf3d165b17113d841d8c0ccd33.png

If the observer in S sees an object moving along the x axis at velocity w , then the observer in the S' system, a frame of reference moving at velocity v in the x direction with respect to S , will see the object moving with velocity w' where

圖片參考：http://upload.wikimedia.org/math/2/9/1/291bfd8042576ef4c34fb191693e72c0.png
This equation can be derived from the space and time transformations above.

圖片參考：http://upload.wikimedia.org/math/c/1/7/c17e48620b3b383bfab22d44ee0515cd.png

In special relativity, Newton's second law does not hold in its form F = ma, but it does if it is expressed as

圖片參考：http://upload.wikimedia.org/math/1/9/1/191b599be1321938d227a606af349343.png
) and "m" is the invariant mass. Thus, the force is given by

圖片參考：http://upload.wikimedia.org/math/2/6/6/26626e99f5f79c99658b3a1ce2796116.png
Carrying out the derivatives gives

圖片參考：http://upload.wikimedia.org/math/8/a/2/8a2c123690501b42acdf62b19f694027.png
, can also be expressed as

圖片參考：http://upload.wikimedia.org/math/4/3/8/43812007d51698d3c1399577a3f3dc26.png
If the acceleration is separated into the part parallel to the velocity and the part perpendicular to it, one gets

圖片參考：http://upload.wikimedia.org/math/7/1/a/71a2ba9081a1c15f325c8cd4462b8ade.png