How does one get time dilation, length contraction, and E=mc^2 from spacetime metric?

[RelativeQuant]

How does one get time dilation, length contraction, and E=mc^2 from the spacetime metric?

Suppose all that you are given is x₁² + x₂² + x₃² - c²t² = s²

How do you derive time dilation, length contraction, and E=mc^2 from this?

What is the most direct way to do this?

 

[Vanadium 50]

You should take a look at Spacetime Physics, by Taylor and Wheeler, where this is all covered. It's only about 100 pages.

 

[RelativeQuant]

Thanks! Is there a quick answer of any resources on the web? Thanks! :)

 

[RelativeQuant]

OK I found a concise way to derive time dilation & length contraction from the spacetime metric:
http://en.wikipedia.org/wiki/Introd...contractions:_more_on_Lorentz_transformations

Does anyone know how to derive E=mc^2 from time dilation and/or length contraction? Or from the spacetime metric?

Thanks! :)

 

[Nugatory]

How do you derive time dilation, length contraction, and E=mc^2 from this?
What is the most direct way to do this?

Start with time dilation. The squared distance between (0,0,0,0) and (T,vT,0,0) must equal the squared distance between (0,0,0,0) and (T',0,0,0) because it's the same interval between the same two points, namely the endpoints of a journey taken at speed $v$ as viewed by the traveller (primed coordinates) and an observer moving at speed $v$ relative to the traveller (unprimed coordinates). Use the metric to calculate the squared distances, equate them, and solve for the ratio of T' to T.

 

[PAllen]

OK I found a concise way to derive time dilation & length contraction from the spacetime metric:
http://en.wikipedia.org/wiki/Introd...contractions:_more_on_Lorentz_transformations

Does anyone know how to derive E=mc^2 from time dilation and/or length contraction? Or from the spacetime metric?

Thanks! :)

This is not a simple. I don't think it can really be derived from the metric alone. You need to bring in, for example, energy and momentum conservation. Then, what you end up deriving from this plus the metric is E² - p²c² = m²c⁴ (once you have the right prior motivation, this just says that the 4-norm of m * 4-velocity unit vector is m, which is trivially true). When momentum is zero, you have the special case E=mc².