Relationship Between Spatial Expansion and Gravity's Force?

[jpescarcega]

Fellow Nerds,
I'm looking for a quantitative relationship between the gravitational strength of a point on a field and the speed of expansion of space at that point. Given a cosmological constant and a metric, is it possible to pinpoint a certain point of space and ask how quickly that space is expanding?
I'm assuming that space doesn't just rapidly/rigidly begin expanding in a vacuum where "gravity equals zero". In GR, gravity extends forever and never reaches zero, and if the cosmological constant is supposed to represent an antigravity force (dark energy), that would suggest that space expands at different rates, since that force isn't just immediately overtaken -- gravity can be weak enough at a certain point to only counter half of the antigravity force, right?
I'm not asking how to get hubble's constant. The FLRW solution doesn't seem to answer the question, nor do the Friedmann equations. I'm talking about looking at a gravitational field (where the cosmological constant is taken into account in the Einstein Field Equations) and picking a point on that field, (where you know the strength of gravity at that point) and asking: "what is the speed of the expansion of space at that point?" (if it is expanding).
Excited To Read Your Answers,
-JP

 

[pervect]

The Friedmann equations, https://en.wikipedia.org/wiki/Friedmann_equations, are probably the closest thing to what you're looking for. But you won't find "gravitational strength" in them anywhere.

I would interpret "gravitational strength" as applying more logically to the second Friedmann equation, which I would describe informally as "gravitational attraction causes the expansion to deaccelerate". Note that in this formulation, you can have expansion or contraction independently of the "gravitational field strength", the "gravitational field strength" doesn't control expansion or contraction, it controls whether or not the expansion is accelerating, deaccelerating, or remaining constant.

But it's not really clear what the term means, so it's possible I'm thinking of "gravitational field strength" differently than you. Additionally, other posters might interpret it differently and come up with a different-sounding interpretation of the same equations. So it's hard to be unambiguous without getting into a lot of technicalities. I'm not sure why you don't regard predicting the Hubble constant as a measure of expansion either.

 

[Dale]

I'm not asking how to get hubble's constant. The FLRW solution doesn't seem to answer the question, nor do the Friedmann equations.

Then I suspect that your question is not well formed, since those are the relevant equations.

Can you pose your question in terms of the metric or the curvature or even the Christoffel symbols?

 

[jpescarcega]

Then I suspect that your question is not well formed, since those are the relevant equations.

Can you pose your question in terms of the metric or the curvature or even the Christoffel symbols?

The simplest solution I can think of to put this problem in is a Schwarzschild-DeSitter Solution, https://en.wikipedia.org/wiki/De_Sitter–Schwarzschild_metric. At a point in Schwarzschild-DeSitter space, at a certain moment, space is expanding at a speed. That point of space is subject to the gravity of the object being described in Sz-DS Space, yet is still expanding. My question, then, is whether there is a mathematical equation/relationship between the gravitational field strength at that point, and the speed at which that point expands (and/or the acceleration of that expansion).

 

[jpescarcega]

I would interpret "gravitational strength" as applying more logically to the second Friedmann equation, which I would describe informally as "gravitational attraction causes the expansion to deaccelerate". Note that in this formulation, you can have expansion or contraction independently of the "gravitational field strength", the "gravitational field strength" doesn't control expansion or contraction, it controls whether or not the expansion is accelerating, deaccelerating, or remaining constant.

The gravitational field strength at that point, then, determines the rate of acceleration of expansion?

But it's not really clear what the term means, so it's possible I'm thinking of "gravitational field strength" differently than you.

Hopefully this rewording clears that up: by the "gravitational strength of a point" I mean the strength of the gravitational field at that point.

I'm not sure why you don't regard predicting the Hubble constant as a measure of expansion either.

While it's a measure of expansion, I'm not so sure that it's as specific as I'd like it to be. It's a generalization, because 74.3 Km/Sec/Mpc doesn't really tell me the acceleration of a point that is, say, 72 Km away from the center of mass, (or does it?). More as a hunch, I feel that since different gravitational fields may have different gravitational strengths at the same distance away from the center of mass, that it doesn't make sense that that same point away from the center should have the same rate of expansion (or acceleration) in both fields, which would seem to be the implication when solely using the Hubble constant as a measure of expansion.

 

[Dale]

At a point in Schwarzschild-DeSitter space, at a certain moment, space is expanding at a speed.

How do you get that from the metric?

 

[jpescarcega]

How do you get that from the metric?

Schwarzschild DeSitter Space is the Schwarzschild Solution with a positive cosmological constant.
The metric is the same as the Schwarzschild Metric.

 

[pervect]

The gravitational field strength at that point, then, determines the rate of acceleration of expansion?

Hopefully this rewording clears that up: by the "gravitational strength of a point" I mean the strength of the gravitational field at that point.

Sorry, that doesn't help much.

Let's try this. If the density rho, and the pressure P and the cosmological constant $\Lambda$ are all zero, so that the Riemann tensor is zero, would you agree that there is "no gravity", and we might say "the gravitational field strength" is zero?

While we still don't have a full definition of "gravitational field strength", we at least know when it is zero. We'll call the case where it is zero a "gravity free universe". We can also say that such a gravity-free universe can either be expanding or not expanding, depending on our choice of coordinates.

See for instance the Milne metric, https://en.wikipedia.org/wiki/Milne_model

I'll just point out the highlights. I feel there may be some confusion regarding the terminiology, but it would be even more confusing to attempt to straighten it out at this point fully. Rather I'll focus on the parts important to the argument. For a "gravity free" universe, we can have a(t) = 1 and a Euclidean spatial geometry, or we can have a(t) = t and a hyperbolic spatial geometry. In either case, $d a(t)/ dt$ is constant, so that $d^2 a(t)/dt^2 = 0$. Hence we can say that in a gravity-free universe, the expansion does not accelerate.

 

[PeterDonis]

Schwarzschild DeSitter Space is the Schwarzschild Solution with a positive cosmological constant.
The metric is the same as the Schwarzschild Metric.

This doesn't answer DaleSpam's question. What he is asking is, what specific quantities lead you to say that "space is expanding at a speed"?

 

[jpescarcega]

what specific quantities lead you to say that "space is expanding at a speed"?

I'm not so certain that specific quantities are needed. Expansion of space is accelerating, which is established through positive cosmological constant. At a certain moment in time, that space will be expanding at a certain speed. Acceleration and time multiplied together give a velocity.

 

[PeterDonis]

I'm not so certain that specific quantities are needed.

Unless you can give some, I have no idea what your statement "space is expanding at a speed" means, physically. And if you can't give specific quantities, you should consider the possibility that it's because the statement is in fact meaningless, physically.

Expansion of space is accelerating, which is established through positive cosmological constant.

"Accelerating" here does not mean what you think it means. See below.

At a certain moment in time, that space will be expanding at a certain speed.

No; at a certain moment of coordinate time in a particular system of coordinates, two "comoving" objects will be moving apart at a coordinate speed which depends on the distance between them. There is no single speed that is the "speed at which space is expanding", and there is no single rate of change that is the "acceleration of space". Also, the coordinate speeds at which comoving objects move apart are not directly measurable; they are constructions in the standard cosmological model we use.

Acceleration and time multiplied together give a velocity.

No; acceleration integrated over a period of time gives a change in velocity. But even with that correction, again, "acceleration" here doesn't mean what you think it means. See above.

 

[jpescarcega]

Unless you can give some, I have no idea what your statement "space is expanding at a speed" means, physically. And if you can't give specific quantities, you should consider the possibility that it's because the statement is in fact meaningless, physically.

Could you elaborate on being physically meaningless?

No; at a certain moment of coordinate time in a particular system of coordinates, two "comoving" objects will be moving apart at a coordinate speed which depends on the distance between them. There is no single speed that is the "speed at which space is expanding", and there is no single rate of change that is the "acceleration of space". Also, the coordinate speeds at which comoving objects move apart are not directly measurable; they are constructions in the standard cosmological model we use.

Doesn't space have to expand at some specific rate, though? For example, one patch of space expands into three patches of space each second. Versus one patch of space expanding into two patches each second. The impacts in both scenarios are different. If we can measure how quickly galaxies are moving apart from us over a certain amount of time, we can get an acceleration, and isn't that measurement mostly a reflection of the expansion of space in between galaxies... Hubble's Constant?
Assuming that space is expanding, and assuming that takes an amount of time, can't a velocity be taken out of that?
(Thank you for your responses, also)

 

[PeterDonis]

Could you elaborate on being physically meaningless?

A statement is physically meaningless if it does not correspond to anything that can be observed, and does not correspond to anything that can be interpreted in terms of a physical theory or model. In your case, "the speed at which space is expanding" is not an observable and does not correspond to anything in the cosmological model of the universe.

Doesn't space have to expand at some specific rate, though?

No.

For example, one patch of space expands into three patches of space each second.

What is a "patch of space"?

If we can measure how quickly galaxies are moving apart from us over a certain amount of time, we can get an acceleration

Yes, but this will be an acceleration of the galaxies relative to us (with some caveats), not an acceleration of "space".

isn't that measurement mostly a reflection of the expansion of space in between galaxies

No.

Hubble's Constant?

What are the units of Hubble's constant? Are they the units of acceleration? Or speed? Take a look and see.

 

[Dale]

Schwarzschild DeSitter Space is the Schwarzschild Solution with a positive cosmological constant.
The metric is the same as the Schwarzschild Metric.

Sorry I wasn't clear about my question. Let me try again.

At a point in Schwarzschild-DeSitter space, at a certain moment, space is expanding at a speed.

What is it about the metric that makes you claim that space is expanding at a speed? I.E. what quantities are you deriving from the metric that would correspond to the speed that space is expanding?