Exploring Einstein's Field Equations: Space Density

[sqljunkey]

Hi,

I'm trying to understand Einstein's field equations conceptually, does it describe space density in a region of space by any chance? Like there is more space in this region compared to this other region. Thanks.

 

[David Lewis]

The left side of the equation expresses the curvature of spacetime.

 

[timmdeeg]

... And the right side of the equation expresses the sources of gravity.

 

[David Lewis]

Yes. The right side is the energy and momentum in the region under consideration.

 

[PeterDonis]

does it describe space density in a region of space by any chance? Like there is more space in this region compared to this other region

The concept of "space density" that you describe isn't well-defined, so no.

 

[Ibix]

I'm trying to understand Einstein's field equations conceptually, does it describe space density in a region of space by any chance? Like there is more space in this region compared to this other region.

Not really. I take it you've seen the kind of picture that shows a 3d cubic grid distorted towards a planet? Unfortunately that's a very limited picture, and it's quite difficult to work out what "density of space" would mean.

Einstein's field equations describe curvature of spacetime. I don't think there's a really good visualisation in general, but here's an idea of where you (and the distorted grids) are going wrong.

Here is Flamm's paraboloid: https://en.m.wikipedia.org/wiki/Schwarzschild_metric#/media/File%3AFlamm.jpg This is an embedding of a part of spacetime (the 2d slice of space, as defined by a Schwarzschild observer, extending outwards from the equator of a Schwarzschild black hole) in flat 3d space. It shows some aspects of the intrinsic curvature of spacetime as (more easily visualised) extrinsic curvature. Imagine drawing concentric rings on that paraboloid so that the distance in the surface between each ring and the next is the same. Now look straight down at the paraboloid from above. The rings near the middle will be closer together (more densely packed) than rings further out because more of the distance between them is in the out-of-the-page direction that has been surpressed by the viewing angle. Does this mean that "space is more dense near mass", or does it mean that the view of Flamm's paraboloid has lost some of the important aspects?

General relativity is all about curved manifolds. "Density of space" doesn’t come into it.

 

[Jonathan Scott]

I'd say there is a sense in which describing space as "more dense" in the region of a mass is reasonable, although one also has to include the fact that time varies in a similar way.

Space-time itself is shaped by gravity, but as the clocks and rulers of local observers follow time and space, the shape on a large scale can only be described by making a map of it relative to a chosen coordinate system, which means that the apparent shape is to some extent determined by the choice of coordinate system.

For describing the space where the primary gravitational effect is that of a static single central mass, a common practical convention (used for example for General Relativity calculations for the motion of space probes within the solar system) is to use what are called isotropic coordinates, where the scale factor between a local ruler and coordinate space is the same in all directions. In this case, provided that we are not talking about being close to a neutron star or black hole, we can assume to high accuracy that gravity effectively causes clocks at a distance $r$ from mass $m$ to run at a rate which is a tiny fraction $Gm/rc^2$ slower than far away from the mass, and similarly it causes rulers at the same location to shrink by the same fraction. This means that if we scale our coordinate system so that it measures time and space in a way which matches local measurements far away from the mass, then one could say that relative to the coordinate system rulers get a little smaller close to the mass, and hence that in a sense physical space is fractionally more dense there.

If this is combined with the effect of clocks running slower by the same fraction, this has an effect on light as if space had a "refractive index" relative to the coordinate system of approximately $1+2Gm/rc^2$, which causes twice the deflection that either the time or space effect alone would cause.

This approximate view relative to isotropic coordinates also holds to good accuracy for multiple static sources within the same region provided that the fields are not too strong (e.g. see Carroll "Spacetime and Geometry" equation 7.59) , in that each source effectively causes time to slow and rulers to shrink in its vicinity, so the local time rate and ruler size are both decreased by the total Newtonian potential at the relevant location.

 

[timmdeeg]

For describing the space where the primary gravitational effect is that of a static single central mass, a common practical convention (used for example for General Relativity calculations for the motion of space probes within the solar system) is to use what are called isotropic coordinates, where the scale factor between a local ruler and coordinate space is the same in all directions. In this case, provided that we are not talking about being close to a neutron star or black hole, we can assume to high accuracy that gravity effectively causes clocks at a distance $r$ from mass $m$ to run at a rate which is a tiny fraction $Gm/rc^2$ slower than far away from the mass, and similarly it causes rulers at the same location to shrink by the same fraction.

I'm not familiar with isotropic coordinates, just wonder why the ruler is shrinking. According to equation 11.22 the radial ruler distance exceeds its coordinate distance in Schwarzschild spacetime. So if I see it correctly the ruler distance shrinks if it is moved away from the mass and coincides with its proper length in flat spacetime.

How is the ruler distance measured in your example and shouldn't the result be independent from chosen coordinates if measured locally?

I'm not sure if this reasoning makes sense, so please correct in case not.

 

[Jonathan Scott]

If the radial ruler distance exceeds the coordinate distance, that means that the ruler is effectively shrunk a bit. If you measure something with a shrunk ruler, the result is larger than it would have been with the right size ruler. For a local observer, no shrinkage is observable, as it is only relative to the coordinate map.

In Schwarzschild coordinates only radial distances are shrunk relative to the coordinate map. Tangential distances are by definition the same as local distances. The Schwarzschild coordinate map makes it easier to solve the Schwarzschild equation, but isotropic coordinates (which effectively shift the radial coordinate slightly until the scale factors are the same in all directions) are easier to compare with Newtonian models.

 

[timmdeeg]

If the radial ruler distance exceeds the coordinate distance, that means that the ruler is effectively shrunk a bit. If you measure something with a shrunk ruler, the result is larger than it would have been with the right size ruler.

I understand the second sentence. But perhaps I haven't the correct notion of what ruler distance means. It seems obvious however that the ruler distance equals the coordinate distance if $m=0$. If you say "the ruler is effectively shrunk a bit" how do you define the length of the ruler in this case? Or do you think of tidal forces here?

 

[Jonathan Scott]

I understand the second sentence. But perhaps I haven't the correct notion of what ruler distance means. It seems obvious however that the ruler distance equals the coordinate distance if $m=0$. If you say "the ruler is effectively shrunk a bit" how do you define the length of the ruler in this case? Or do you think of tidal forces here?

The length of a ruler is simply used to illustrate the size of a unit length of local space compared with the chosen coordinate system.

 

[pervect]

I understand the second sentence. But perhaps I haven't the correct notion of what ruler distance means. It seems obvious however that the ruler distance equals the coordinate distance if $m=0$. If you say "the ruler is effectively shrunk a bit" how do you define the length of the ruler in this case? Or do you think of tidal forces here?

When one talks about rulers shrinking, one is basically creating a hypothetical entity that has no direct physical observable basis. One is free to make up whatever hypothetical entites one likes. The goal is to make up such entities in a manner that is useful in helping one to understand the physics of the physical, things one can observe and measure. The physical things one can measure are the SI distance with the modern SI definition based on the light travel time, or any of the other earlier standard definitions of the meter including the prototype meter bar which was one of the earliest defintions, though perhaps not the earliest. The SI defintions have changed over time, but they are thought of as different protocols for measuring the same conceptual entity, which we call the physical distance. By definition, this sort of distance doesn't shrink or change, it just is - because that's how we define it. We need to define it so that we can communicate with one another. When we talk about rulers shrinking, we're making up something that's not the same as physical distance. Sometimes this can be useful, if one has a shared understanding of what one is talking about, even though it's not the standard "physical" SI distance.

Sometimes people who are not familar or who have doubrts about special relativity seem more comfortable with the older standard defintitions of the meter. It's a belief that special relativity actually works and that General relativity is locally equivalent to special relativity in a small enough that links the older definitions of the meter to the earlier ones.

 

[timmdeeg]

By definition, this sort of distance doesn't shrink or change, it just is - because that's how we define it. We need to define it so that we can communicate with one another. When we talk about rulers shrinking, we're making up something that's not the same as physical distance.

Yes, one has to define it. I feel more comfortable to understand a ruler as a physical thing. Then the proper distance between two shells can be measured with a ruler and it follows that the coordinate difference between two shells is increasingly shrinking with decreasing $r$-coordinate relativ to their distance measured by rulers.