Measuring Coordinates in Strong Gravity: Schwarzschild Metric

In summary, the Schwarzschild metric describes an asymptotically flat spacetime where the ##r## coordinate can be interpreted as distance from the center far away from the event horizon. However, as the curvature becomes significant near the event horizon, our common sense of ##r## breaks down. Coordinates are defined, not measured, and can be defined in terms of a relationship to measurements. The definition and measurement of the ##r## coordinate as the integral of proper length along a curve with certain constraints is valid both inside and outside the event horizon. This is usually stated in terms of measuring the area, but with spherical symmetry the two are equivalent.
  • #1
victorvmotti
155
5
We know that Schwarzschild metric describes an asymptotically flat spacetime. This means that far away from the event horizon we can safely interpret the ##r## coordinate as distance from the center.

But when close enough to the event horizon the curvature becomes significant and our common sense of ##r## breaks.

The question is that what is understood as measurement of the coordinates near very strong gravity?
 
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  • #2
Coordinates are defined, not measured. You can of course define them such that they have some defined relationship to measurements. In that case you perform the measurements, apply the defined relationship and obtain the coordinate.
 
  • #3
Is it reasonable to define and "measure" the ##r## coordinate like this:

##r## is in the Schwarzschild metric the integral of the proper length along the curve $$dt = dr = d\theta = 0$$ for ##\phi## from ##0## to ##2\pi## divided by ##2\pi##?

Is this definition and measurement valid inside the event horizon as well?
 
  • #4
Yes. Although usually it is stated in terms of measuring the area, but with spherical symmetry what you wrote is equivalent
 
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FAQ: Measuring Coordinates in Strong Gravity: Schwarzschild Metric

What is the Schwarzschild metric and why is it important in measuring coordinates in strong gravity?

The Schwarzschild metric is a mathematical description of the curvature of spacetime around a non-rotating, spherically symmetric massive object, such as a black hole. It is important in measuring coordinates in strong gravity because it allows us to understand how the geometry of space and time is affected by the presence of a massive object, and how this affects the motion of objects and the passage of time.

How is the Schwarzschild metric used to measure coordinates in strong gravity?

The Schwarzschild metric is used in the field of general relativity to calculate the coordinates of objects in strong gravity, such as black holes. By solving the equations of the metric, we can determine the distance from the center of the massive object, the time dilation, and the escape velocity, among other things.

Can the Schwarzschild metric be applied to other objects besides black holes?

Yes, the Schwarzschild metric can be applied to any non-rotating, spherically symmetric massive object. This includes neutron stars and other compact objects with high densities and strong gravitational fields.

What is the significance of measuring coordinates in strong gravity using the Schwarzschild metric?

Measuring coordinates in strong gravity allows us to understand the behavior of matter and energy in extreme environments, and provides insight into the nature of spacetime and the laws of physics. This has important implications for our understanding of the universe and the phenomena that occur within it.

Are there any limitations to using the Schwarzschild metric in measuring coordinates in strong gravity?

One limitation is that the Schwarzschild metric only applies to non-rotating, spherically symmetric objects. It also does not take into account the effects of quantum mechanics, which may be necessary to fully understand the behavior of matter and energy in strong gravity. Additionally, the Schwarzschild metric breaks down at the center of a black hole, where the curvature of spacetime becomes infinite.

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