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

In summary, the conversation discusses how to derive time dilation, length contraction, and E=mc^2 from the spacetime metric and other related concepts. It is recommended to read Spacetime Physics by Taylor and Wheeler for a more thorough understanding. While there are concise ways to derive time dilation and length contraction, deriving E=mc^2 from the metric alone is not simple and requires additional concepts such as energy and momentum conservation.
  • #1
RelativeQuant
6
0
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?
 
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  • #2
You should take a look at Spacetime Physics, by Taylor and Wheeler, where this is all covered. It's only about 100 pages.
 
  • #3
Thanks! Is there a quick answer of any resources on the web? Thanks! :)
 
  • #5
RelativeQuant said:
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.
 
  • #6
RelativeQuant said:
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.
 
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1. What is spacetime metric and how does it relate to time dilation, length contraction, and E=mc^2?

Spacetime metric is a mathematical concept used in Einstein's theory of relativity to describe the geometry of spacetime. It relates to time dilation, length contraction, and E=mc^2 by providing a way to calculate the effects of gravity and motion on the measurements of time, space, and energy.

2. How does gravity cause time dilation and length contraction?

In general relativity, gravity is described as the curvature of spacetime caused by the presence of massive objects. This curvature affects the passage of time and the lengths of objects, resulting in time dilation and length contraction. The greater the gravitational force, the more significant these effects become.

3. What is the equation for time dilation and length contraction?

The equation for time dilation is t' = t√(1 - v^2/c^2), where t' is the measured time in a moving frame of reference, t is the time in a stationary frame of reference, v is the relative velocity between the two frames, and c is the speed of light. The equation for length contraction is L' = L√(1 - v^2/c^2), where L' is the measured length in a moving frame of reference and L is the length in a stationary frame of reference.

4. How does the spacetime metric equation, ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2, relate to E=mc^2?

The equation for spacetime metric, ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2, is used to calculate the spacetime interval between two events. This interval is a measure of the distance between two points in spacetime. When combined with the energy-momentum equation, E^2 = (pc)^2 + (mc^2)^2, it can be used to derive the famous equation E=mc^2, which relates energy and mass.

5. Can the effects of time dilation and length contraction be observed in everyday life?

Yes, the effects of time dilation and length contraction can be observed in everyday life, although they are usually very small and only noticeable under extreme conditions. For example, the Global Positioning System (GPS) must account for time dilation effects in order to accurately track location and time. Additionally, high-speed particles in particle accelerators experience significant length contraction due to their high velocities.

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