Checking if Schwarzschild Space-Time is Locally Minkowskian

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In summary, the conversation discusses the relationship between tangent spacetime and Minkowski spacetime, specifically in the context of Schwarzschild spacetime. It is explained that all spacetimes are locally Minkowskian and that the tangent space can be obtained through Riemann normal coordinates. It is clarified that the isomorphism between the tangent space and Minkowski spacetime does not involve Lorentz transformations, but rather basic linear algebra.
  • #1
jk22
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hello,

I would like to check that, ie if the tangent spacetime of schwarzschild can be linked to minkowski spacetime via a lorentz transformation.

But i have a problem from the onset since the tangent space seems to me to exist only in exterior geometry. For example the tangent plane to a surface lies in embedding 3d space.

Else how to find the tangent space given only interior coordinates and metric ?

Thanx.
 
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All space-times are locally Minkowskian; just setup Riemann normal coordinates in an appropriate neighborhood of an event. This can be done generally for any space-time, not just Schwarzschild. You can't link the tangent space to space-time at an event to Minkowski space-time via a Lorentz transformation; I'm not sure where you even got that idea. The isomorphism between the tangent space at any event and Minkowski space-time can be obtained explicitly using basic linear algebra but it has nothing to do with Lorentz transformations.
 

FAQ: Checking if Schwarzschild Space-Time is Locally Minkowskian

What is Schwarzschild space-time?

Schwarzschild space-time is a mathematical model that describes the curvature of space and time around a non-rotating mass, such as a star or planet. It was first proposed by German physicist Karl Schwarzschild in 1916 and is a key concept in Einstein's theory of general relativity.

How do we check if Schwarzschild space-time is locally Minkowskian?

In order to check if Schwarzschild space-time is locally Minkowskian, we need to perform a coordinate transformation to a new set of coordinates that are locally flat. This is known as the "locally Minkowskian frame" or "locally inertial frame". If the transformed space-time is flat, then Schwarzschild space-time is locally Minkowskian.

Why is it important to determine if Schwarzschild space-time is locally Minkowskian?

Determining if Schwarzschild space-time is locally Minkowskian is important because it helps us understand the nature of gravity and the effects of massive objects on the curvature of space and time. It also allows us to make accurate predictions about the behavior of objects in the presence of such massive objects.

What are the implications if Schwarzschild space-time is not locally Minkowskian?

If Schwarzschild space-time is not locally Minkowskian, then this would indicate that Einstein's theory of general relativity may not accurately describe the behavior of massive objects and gravity. This could also have implications for our understanding of the universe and the laws of physics.

How is the local Minkowskian frame used in practical applications?

The local Minkowskian frame is used in practical applications, such as in GPS systems, where precise measurements and predictions are needed in the presence of massive objects, like Earth. It also helps in understanding the behavior of black holes and other massive objects in space.

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