Validating a ds^2 Metric in General Relativity

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In summary, the conversation discusses the validity of a metric in theories involving general relativity. It is noted that the metric alone is not enough and candidate solutions must also include a manifold, a metric, and a stress-energy tensor field. The Einstein Curvature Tensor is then computed and compared to the stress-energy tensor. The validity of the stress-energy tensor is questioned and it is suggested to refer to Geroch's viewpoint for further understanding.
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NotASmurf
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Hey all, in every theory that involves GR you see they give their space-time metric, but very few show any other math related to it, how does one know if a metric is valid?
 
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The metric alone is not sufficient.
Conceptually, candidate solutions of the field equations have the general form (M,g,T)...
you have to spell out the manifold (set of events) M,
the metric g [as you have provided],
and a stress-energy tensor field T (describing the matter distribution... special case: in vacuum, T=0).

Then you substitute into field equations:
from the metric tensor g, compute the Einstein Curvature Tensor G... and ask if it is equal (up to conventional constants) to the stress-energy T.

You could just work out the Einstein tensor G and declare your T to be equal to it...
However,
physically, we have to ask questions about the matter distribution T...
is this T physically reasonable? (Does it satisfy certain energy conditions, etc..? Is this realizable by real matter? or some weird exotic matter?)

Here is the source of this viewpoint: Geroch, General Relativity from A to B, p 172
https://books.google.com/books?id=AC1OCgAAQBAJ&pg=PA172&lpg=PA172
 
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Thanks, :)
 

Related to Validating a ds^2 Metric in General Relativity

1. What is a ds^2 metric in general relativity?

A ds^2 metric is a mathematical expression that describes the geometry of spacetime in general relativity. It is used to calculate the distance between two points in a curved spacetime, taking into account the effects of gravity.

2. Why is validating a ds^2 metric important in general relativity?

Validating a ds^2 metric is important because it allows us to confirm that the equations and principles of general relativity accurately describe the behavior of objects in the universe. It also helps us to make accurate predictions and understand the effects of gravity on space and time.

3. How is a ds^2 metric validated in general relativity?

A ds^2 metric is validated by comparing its predictions to observational data and experimental results. This involves testing the metric in different scenarios and making sure that it accurately describes the observed behavior of objects in the universe.

4. What are some challenges in validating a ds^2 metric in general relativity?

One of the main challenges in validating a ds^2 metric is the complexity of the equations involved. General relativity is a highly complex theory, and it can be difficult to accurately test and validate the metric in all possible scenarios. Additionally, obtaining precise observational data can also be a challenge.

5. How does validating a ds^2 metric contribute to our understanding of the universe?

Validating a ds^2 metric is essential for our understanding of the universe because it helps to confirm the validity of general relativity as a theory of gravity. It also allows us to make accurate predictions and explore the behavior of objects in extreme conditions, such as near black holes or during the early stages of the universe.

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