Schwarzschild metric as induced metric

In summary, according to the Nash theorem, every Riemannian manifold can be isometrically embedded into some Euclidean space. It is also possible to find a submanifold in pseudoeuclidean space with a Schwarzschild metric, with 6 dimensions being the minimum required. Chris Clarke's work has shown that every 4-dimensional spacetime can be embedded isometrically in higher dimensional flat space, with 90 dimensions being sufficient. However, a particular spacetime may be embeddable in a flat space with less than 90 dimensions. Clarke's work also suggests that all GR solutions can be locally embedded in 10 dimensions.
  • #1
paweld
255
0
According to Nash theorem http://en.wikipedia.org/wiki/Nash_embedding_theorem" every Riemannian manifold can be isometrically embedded
into some Euclidean space. I wonder if it's true also
in case of pseudoremanninan manifolds. In particular is it possible to find
a submanifold in pseudoeuclidean space that, the metric induced on it will be
Schwarzschild metric? How many dimensions we need?
 
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  • #2
Chris Clarke* showed that every 4-dimensional spacetime can be embedded isometically in higher dimensional flat space, and that 90 dimensions suffices - 87 spacelike and 3 timelike. A particular spacetime may be embeddable in a flat space that has dimension less than 90, but 90 guarantees the result for all possible spacetimes.

* Clarke, C. J. S., "On the global isometric embedding of pseudo-Riemannian
manifolds," Proc. Roy. Soc. A314 (1970) 417-428
 
  • #3
You need 6 dimensions to embed a Schwarzschild solution. I think that all GR solutions can be (locally) embedded in 10 dimensions.
 
  • #4
Passionflower said:
You need 6 dimensions to embed a Schwarzschild solution.
How do you know it?
 

FAQ: Schwarzschild metric as induced metric

What is the Schwarzschild metric as an induced metric?

The Schwarzschild metric is a mathematical description of the curvature of space-time around a non-rotating and uncharged massive object, such as a star or a black hole. It is known as an "induced metric" because it is derived from a more general metric, known as the Kerr metric, which describes the curvature of space-time around a rotating massive object.

How is the Schwarzschild metric derived?

The Schwarzschild metric is derived from the Einstein field equations, which are a set of equations that describe the relationship between the curvature of space-time and the distribution of matter and energy within it. Specifically, the Schwarzschild metric is a solution to the Einstein field equations in a vacuum, meaning that there is no matter or energy present in the region of space being described.

What does the Schwarzschild metric tell us about space-time?

The Schwarzschild metric tells us that the curvature of space-time is not fixed, but rather depends on the distribution of matter and energy within it. In the case of a non-rotating and uncharged massive object, the curvature of space-time is described by a simple, spherical shape. However, for more complex objects, such as rotating or charged objects, the curvature of space-time can be much more complicated.

What is the significance of the Schwarzschild radius in the Schwarzschild metric?

The Schwarzschild radius is a specific distance from the center of a massive object, beyond which the escape velocity exceeds the speed of light. This radius is a fundamental concept in the Schwarzschild metric, as it represents the boundary of the event horizon of a black hole. Objects that pass through this radius are unable to escape the gravitational pull of the black hole and are therefore trapped within it.

Can the Schwarzschild metric be tested or observed?

Yes, the Schwarzschild metric has been tested and confirmed through various experiments and observations, such as the bending of light around massive objects and the slowing of time near massive objects. However, it is important to note that the Schwarzschild metric is a mathematical description and cannot be directly observed. It is a useful tool for understanding the behavior of space-time around massive objects, but it is not a physical object that can be observed in the traditional sense.

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