Actual examples of spacetime curvature

[geordief]

Is there a layman's guide to  the mathematics of calculating the spacetime curvature of a body of mass M inhabiting a flat space?

 

Suppose we take the mathematics of calculating the curvature at any point as given (it's not ,of course ;I don't understand that yet except that it seems to be a tensor) how does one ,in broad terms proceed from there to describe the curvature of the body on its surface ,in its interior and extending beyond?

Does one integrate all the individual  points of curvature?

[Strange]

Not sure it is what you are looking for, but this is a really good overview of what the equations of GR mean, and what the consequences are (in relatively simple mathematical and physical terms) for some special cases: http://math.ucr.edu/home/baez/einstein/

[studiot]

32 minutes ago, geordief said:

Is there a layman's guide to  the mathematics of calculating the spacetime curvature of a body of mass M inhabiting a flat space?

 

Suppose we take the mathematics of calculating the curvature at any point as given (it's not ,of course ;I don't understand that yet except that it seems to be a tensor) how does one ,in broad terms proceed from there to describe the curvature of the body on its surface ,in its interior and extending beyond?

Does one integrate all the individual  points of curvature?

Do you realise you require a mass density distribution to do this, not "a body of mass M"  and why ?

[Strange]

57 minutes ago, studiot said:

Do you realise you require a mass density distribution to do this, not "a body of mass M"  and why ?

Could you explain why you think a "a body of mass M" is not a "mass density distribution".

And can you also explain why this is required.

That would be much more helpful. Thank you.

 

[Markus Hanke]

1 hour ago, geordief said:

Is there a layman's guide to  the mathematics of calculating the spacetime curvature of a body of mass M inhabiting a flat space?

Not that I am aware of, since the calculation requires in-depth knowledge of how to handle systems of non-linear partial differential equations, which goes far beyond what most amateurs would be familiar with. There are of course textbooks that explicitly go through this, but none of them is aimed at amateurs (they are usually at post grad level).

1 hour ago, geordief said:

Suppose we take the mathematics of calculating the curvature at any point as given (it's not ,of course ;I don't understand that yet except that it seems to be a tensor) how does one ,in broad terms proceed from there to describe the curvature of the body on its surface ,in its interior and extending beyond?

Normally, spacetime curvature would not be a given quantity, you need to find it first. In order to do so, you have to first solve the Einstein equations for the physical scenario in question; and the quantity you solve it for is the metric (more accurately - the components of the metric tensor). Once you know the metric, you can then calculate spacetime curvature with it, which is - in the most general case - described by the Riemann curvature tensor.

To find the metric outside a body, you solve the vacuum equations

\[R_{\mu \nu}=0\]

For the metric in the interior of the body, you need to find a solution for the full Einstein equations

\[R_{\mu \nu } -\frac{1}{2} g_{\mu \nu } R=\kappa T_{\mu \nu }\]

where the stress-energy tensor on the right describes the distribution of energy-momentum inside the body. You then match these two solutions at the boundary, i.e. you ensure that the metric remains smooth and continuous at the body’s surface, by appropriate choices of integration constants. This will essentially give you one metric that covers the entire spacetime, interior and vacuum. The Riemann tensor then follows from this accordingly.

Edited February 21, 2020 by Markus Hanke

[swansont]

!

Moderator Note

Not sure why this was in the Lounge. Moved to relativity 

 

[Mordred]

23 hours ago, Markus Hanke said:

Not that I am aware of, since the calculation requires in-depth knowledge of how to handle systems of non-linear partial differential equations, which goes far beyond what most amateurs would be familiar with. There are of course textbooks that explicitly go through this, but none of them is aimed at amateurs (they are usually at post grad level).

Normally, spacetime curvature would not be a given quantity, you need to find it first. In order to do so, you have to first solve the Einstein equations for the physical scenario in question; and the quantity you solve it for is the metric (more accurately - the components of the metric tensor). Once you know the metric, you can then calculate spacetime curvature with it, which is - in the most general case - described by the Riemann curvature tensor.

To find the metric outside a body, you solve the vacuum equations

 

Rμν=0

 

For the metric in the interior of the body, you need to find a solution for the full Einstein equations

 

Rμν−12gμνR=κTμν

 

where the stress-energy tensor on the right describes the distribution of energy-momentum inside the body. You then match these two solutions at the boundary, i.e. you ensure that the metric remains smooth and continuous at the body’s surface, by appropriate choices of integration constants. This will essentially give you one metric that covers the entire spacetime, interior and vacuum. The Riemann tensor then follows from this accordingly.

Nice detailed answer +1