B Mass of Space (B but approaching I)

[Thkaal]

TL;DR Summary

What is the mass/energy of a unit of true vacuum?

Since space is expanding, and that expansion moves things, then those moved things must be moved by something. If moved by space itself, then space must be providing the energy. And if gravity is a function of space and time interacting, then what is the energy or mass of a volume of space that has no matter in it?

 

[PeroK]

An estimate of the vacuum energy density of free space is given here ($10^{-9}$ joules per cubic meter).

https://en.wikipedia.org/wiki/Vacuum_energy

 

[Drakkith]

Since space is expanding, and that expansion moves things, then those moved things must be moved by something.

This is not quite true. The distance between unbound objects (like galaxy clusters) is increasing, yes, but not necessarily because they are being moved by something. It's actually a consequence of geometry in non-flat spacetime. That is, they aren't being moved apart, they are simply moving apart.

Another way to say it is that there is no force being applied to move objects apart, they are simply moving through a certain type of non-flat geometry that's well beyond my full understanding and ability to explain.

And if gravity is a function of space and time interacting, then what is the energy or mass of a volume of space that has no matter in it?

Gravity is the apparent attraction of objects that are moving through curved spacetime. It's not because space and time are interacting (they aren't separate objects), but because the geometry of spacetime is 'bent' in such a way as to make objects that are moving in straight lines in 4D spacetime move in curved lines in 3D space.

If you have trouble visualizing this, just think of the following:
You and a friend stand on the equator of the Earth, facing north, 100 feet apart. You both start walking north. Over time you will find that the distance between the two of you is decreasing, yet neither of you has experienced a force that would move you left or right or turn your slightly. Both of you would say that you've been walking a straight line the whole time. Yet, if you both keep walking for long enough, you will collide with each other at the north pole.

Your observation that you've both walked straight lines is correct. You both walked what is the equivalent of a straight line on a positively curved surface, something which is called a geodesic. Geodesics are paths of shortest distance between two points in a curved geometry, such as the 2D surface of the Earth or 4D spacetime.

 

[Thkaal]

Okay, the two answers I got, one from two different people, contradict each other.

If space is expanding, the space that was 1 cubic meter then becomes 8. Those ~5GeV now become ~40GeV. So, why is that energy not what is moving the objects? That energy just came out of nowhere.

Or, is that energy the "heat" of doing the work of stretching that space?

OH OH OH! Another related question considering stretching space. Isn't that Planck unity of space supposed to be the smallest unit? If it is, where does the new space fit in? Or is this one of those "What's north of the north pole?" (but anyone who understands 5d space can answer that)

 

[PeroK]

Okay, the two answers I got, one from two different people, contradict each other.

I'm not sure that @Drakkith is contradicting the Wikipedia page (which was my answer).

If space is expanding, the space that was 1 cubic meter then becomes 8. Those ~5GeV now become ~40GeV. So, why is that energy not what is moving the objects? That energy just came out of nowhere.

The conservation of energy ultimately stems from the concept of time invariance, which no longer holds in an expanding universe. There's no obligation to have global energy conservation in an expanding space. See, for example:

https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

Or, is that energy the "heat" of doing the work of stretching that space?

It's closer to something like negative pressure.

OH OH OH! Another related question considering stretching space. Isn't that Planck unity of space supposed to be the smallest unit?

No. That's a common misconception. There is no viable model of spacetime where space and time have smallest units. Spacetime (in the current theories) is a continuum (differentiable manifold).

If it is, where does the new space fit in?

The expansion of space is part of the model of spacetime. As time passes, distant objects are further apart; and the total vacuum energy increase. That is a valid solution of the Einstein Field Equations and the Friedmann equation that govern the large scale behaviour of the universe.

Energy is conserved locally (which is what you learn in Physics 101), but globally things are not so simple.

 

[Vanadium 50]

Okay, the two answers I got, one from two different people, contradict each other.

No, no, they don't. The most likely outcome of saying that they do is to get everybody mad at you.

You asked a question based on a faulty premise. One person answered it and one pointed out that the premiise was faulty. What more could you ask for?

OH OH OH!

Great Arnold Horshack impression! But if you have a different question, you can always ask it in its own thread.

 

[PeterDonis]

If space is expanding, the space that was 1 cubic meter then becomes 8.

For a particular choice of coordinates, yes. But coordinate-dependent things don't contain any actual physics.

Those ~5GeV now become ~40GeV.

No. The "total" you are using here is a coordinate-dependent thing. It doesn't contain any actual physics.

Why is that energy not what is moving the objects?

The objects don't feel any force. They are in free fall. There is nothing "moving" them. They are just freely falling through spacetime. The dark energy you are referring to is part of what is determining the geometry of the spacetime they are freely falling through. That's all.

That energy just came out of nowhere.

No, it didn't. The density of dark energy is the same everywhere. No dark energy is created or destroyed anywhere. In mathematical terms, the covariant divergence of the dark energy stress-energy tensor is zero everywhere.

It is a common pop science description to say that "space is expanding" and that this expansion must "create" dark energy where there was none before, but there is no actual invariant anywhere in the math that says that. It's just a pop science claim based on coordinate-dependent things.

Or, is that energy the "heat" of doing the work of stretching that space?

There is no work being done. "Space" is not something you have to do work to "stretch".

Isn't that Planck unity of space supposed to be the smallest unit?

Not in our best current models, no. There are speculations in quantum gravity along these lines, but they are just speculations at this point. Nobody has any experimentally tested theory that says this.

 

[Drakkith]

If space is expanding, the space that was 1 cubic meter then becomes 8. Those ~5GeV now become ~40GeV. So, why is that energy not what is moving the objects? That energy just came out of nowhere.

You are visualizing the expansion of space as 'space itself' expanding and getting larger. Like if have a 1x1x1 unit cube of empty space, this cube expands over time to be 2x2x2, then 3x3x3. This is not correct as far as I know. Expansion is about physical objects moving apart over time, not about space itself somehow expanding. That is, over time unbound objects will simply move apart without any 'new' space being created and inserted between them.

In an infinite universe (which is how we model it) we don't need to create new space or have a physical boundary move outwards to get more room for things to expand into. The universe is infinite, so we already have an infinite amount of space. It's very counterintuitive, I know.

 

[phinds]

That is, over time unbound objects will simply move apart without any 'new' space being created and inserted between them.

And yet, the amount of space between them does increase, which is what causes the confusion.

@Thkaal a better way of thinking about it is that space-time is geometry and things get farther apart because the metric is changing. That is why it is called "metric expansion".

The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time.[1] It is an intrinsic expansion whereby the scale of space itself changes. The universe does not expand "into" anything and does not require space to exist "outside" it. This expansion involves neither space nor objects in space "moving" in a traditional sense, but rather it is the metric (which governs the size and geometry of spacetime itself) that changes in scale.

 

[phinds]

Isn't that Planck unity of space supposed to be the smallest unit?

As has been pointed out, no, it is not. What it IS, according to the current understanding of quantum mechanics, is the smallest distance to which anything can be measured. That doesn't mean things can't be smaller, including distances, it just means that we don't know how to measure anything to any greater resolution than one Planck length because of the way light works with the definition of the Planck length.

Oh, and I should add, that's all just theoretical anyway since modern technology can't measure anything to within a large number of orders of magnitude greater than one Planck length.

 

[Vanadium 50]

I think it was a mistake to allow the OP to derail his own thread, and it's a mistake to ascribe any mysticism to the Planck units. The Planck resistance is 30Ω. There are resistances above and below this number.

No woo required.

 

[berkeman]

I think it was a mistake to allow the OP to derail his own thread

I'm getting dizzy...

 

[Drakkith]

And yet, the amount of space between them does increase, which is what causes the confusion.

Which is why I like to say that the distance between them increases. MUCH less confusion.

 

[PeroK]

You are visualizing the expansion of space as 'space itself' expanding and getting larger. Like if have a 1x1x1 unit cube of empty space, this cube expands over time to be 2x2x2, then 3x3x3. This is not correct as far as I know.

That is what's happening.

Expansion is about physical objects moving apart over time, not about space itself somehow expanding.

The objects move apart because space has expanded.

That is, over time unbound objects will simply move apart without any 'new' space being created and inserted between them.

Space creation is not usually the term that's used. But, in comoving coordinates, there is increasingly more space (and more vacuum energy) over time.

This is perhaps where relying solely on the coordinate-dependent "expansion of space" model starts to confuse the issue somewhat. In any case, the equations that govern the universe at the cosmological scale do not admit a static solution.

In an infinite universe (which is how we model it) we don't need to create new space or have a physical boundary move outwards to get more room for things to expand into. The universe is infinite, so we already have an infinite amount of space. It's very counterintuitive, I know.

I'm not sure this is relevant. The universe might be infinite and it might not be. The metric expansion of space does not rely on an infinite universe. A finite universe must also be expanding (or contracting).

 

[PeterDonis]

The objects move apart because space has expanded.

I don't think this is correct. "Space expanding" is not the cause of anything. It's a coordinate-dependent description of the spacetime geometry. The objects in question are following geodesics so they feel no force, so nothing is "moving them apart".

In terms of spacetime geometry and invariants, what is often referred to as "space expanding" is best described as the congruence of "comoving" worldlines having a positive expansion scalar. But that is a property of the congruence of worldlines, not of "space".

in comoving coordinates, there is increasingly more space (and more vacuum energy) over time.

Yes, but as you note, this is a coordinate-dependent description. As I said earlier, there is no invariant that says "there is more vacuum energy over time". And physics is supposed to be contained in invariants, not coordinate-dependent quantities.

 

[Drakkith]

Is there any significant similarities between objects on a 2d surface moving apart because they are following diverging geodesics vs the same thing happening in spacetime? Are these roughly equivalent to each other minus the obvious of having different numbers of dimensions? Or are they very different?

 

[timmdeeg]

Which is why I like to say that the distance between them increases. MUCH less confusion.

Why is this much less confusion? Distance increases right - cubic distance increases wrong?

 

[PeroK]

Is there any significant similarities between objects on a 2d surface moving apart because they are following diverging geodesics vs the same thing happening in spacetime? Are these roughly equivalent to each other minus the obvious of having different numbers of dimensions? Or are they very different?

The difference is that in GR time is part of the manifold. If we choose comoving time, then the two scenarios are analagous - and we have (in that coordinate system) expanding space. But, comoving time is an arbitrary choice - albeit a very natural choice of coordinates.

There is then the subtle point that once you include time in the spacetime manifold, that manifold itself does not evolve over time. Whereas, in your scenario of the 2D surface expanding we're assuming an absolute time parameter (e.g. as we do in non-relativistic physics).

 

[timmdeeg]

Yes, but as you note, this is a coordinate-dependent description. As I said earlier, there is no invariant that says "there is more vacuum energy over time". And physics is supposed to be contained in invariants, not coordinate-dependent quantities.

That is hard to digest though. Imagine four comoving particles in free fall and lets call space what’s included by them. Do you say that the included space isn’t larger after a while?

Or in other words why do we say distances are increasing but the volume included by them which we hopefully are allowed to call space isn’t?

By the way Peacock in “Cosmological Physics” calls DE as an unlimited reservoir of energy.

 

[PeterDonis]

magine four comoving particles in free fall and lets call space what’s included by them. Do you say that the included space isn’t larger after a while?

If you redefine "space" to mean whatever you want, then of course you can get "space" to mean whatever you want. But this tells you nothing about physics. It only tells you about how you've chosen to redefine words.

why do we say distances are increasing but the volume included by them which we hopefully are allowed to call space isn’t?

Nobody is saying "space", with a particular chosen definition of that word (namely, surfaces of constant coordinate time in standard FRW coordinates), is not "expanding", with a particular chosen definition of that word (namely, that the scale factor in standard FRW coordinates is increasing with coordinate time). We're just pointing out that this is a coordinate dependent concept, and actual physics is contained in invariants.

You can pick out an invariant that can be connected with "space expanding", namely, that the expansion scalar of the congruence of comoving worldlines is positive. But you cannot use that invariant to support any claim about "dark energy being created" or "objects being moved" by this means. Those are the claims that are being argued against here.

 

[PeroK]

That is hard to digest though.

I agree it's hard to digest when you first encounter the idea. Despite all the fuss about Hubble discovering that the universe was expanding, in terms of modern theoretical physics, the expanding universe is a coordinate-dependent concept (although there is an underlying invariant quantity to which it relates).

 

[timmdeeg]

Thanks for your comments @PeterDonis and @PeroK!
Let’s look at a time slice. I guess that then one can’t define a volume included within the four particles in any senseful meaning because the distances between them are spacelike. Would you agree to that? I am just trying to understand what you say within a not too technical limit.

 

[PeterDonis]

Let’s look at a time slice. I guess that then one can’t define a volume included within the four particles in any senseful meaning because the distances between them are spacelike. Would you agree to that?

Not at all. Once you have picked a spacelike slice, you can define 3-volumes within it just like your intuition would suggest (except that if the slice is not flat, you need to allow for the non-Euclidean geometry of the space).

What we are saying is that picking a spacelike slice amounts to picking a coordinate chart, so the 3-volumes you end up defining in that slice are coordinate-dependent.

 

[Ibix]

What we are saying is that picking a spacelike slice amounts to picking a coordinate chart, so the 3-volumes you end up defining in that slice are coordinate-dependent.

@timmdeeg - it's possibly helpful to consider Minkowski spacetime. Draw a Minkowski diagram and add two inertial worldlines. Pick an event on one worldline and draw a horizontal line joining it to the other worldline. That little segment is the 1d "volume" contained between those worldlines, in lower-dimensional analogy to your family of four co-moving points, using that frame.

But I don't need to use that frame - any straight spacelike line connecting that chosen event to somewhere on the other worldline defines a 1d volume according to some inertial frame. And we are not restricted to considering inertial frames, so in fact every curve that passes through the chosen event and is everywhere spacelike while it's between the worldlines defines a possible "volume" for sufficiently outré coordinates.

The same process applies in FLRW spacetime. There are sensible choices of coordinates (any that use cosmological time), and there's so little reason to use any others that people do talk about them as if they were the only choice. But they are not the only choice, and you aren't required to use them. And there are times when it seems sensible to try to avoid making any choice, such as when one is discussing causation.

 

[timmdeeg]

Ok, very helpful, thank you both, @PeterDonis and @Ibix.

So to get rid of the coordinate dependence of a spacelike “volume” we should pick the
“underlying invariant quantity” which @PeroK mentioned in #21.
Can you please elaborate a bit about this quantity. Is it this quantity, perhaps a nameless mathematical expression, which grows with the “size” of the universe? And which then is the underlying thing if we talk about the expanding universe.

 

[PeterDonis]

the
“underlying invariant quantity” which @PeroK mentioned in #21.

I already mentioned it earlier, in post #15.

Can you please elaborate a bit about this quantity.

See post #15.

Is it this quantity, perhaps a nameless mathematical expression, which grows with the “size” of the universe?

No. The closest intuitive concept to the expansion scalar is that it is a rate of expansion. It is not a "size".

And which then is the underlying thing if we talk about the expanding universe.

What do you mean by "the underlying thing"?

 

[timmdeeg]

What do you mean by "the underlying thing"?

the congruence of worldlines ... after re-reading #15 .
A quite abstract thing which for me doesn't have a comprehensible relation to what we call space.

 

[Ibix]

the congruence of worldlines ... after re-reading #15 .
A quite abstract thing which for me doesn't have a comprehensible relation to what we call space.

A congruence of worldlines is a set of worldlines that completely fill all or a part of spacetime, so you can identify any event uniquely by saying "this worldline at its proper time t". All inertial worldlines at mutual rest is a congruence that covers Minkowski spacetime, or the worldlines of comoving observers in FLRW spacetime. You can then put a coordinate free meaning on "are they moving apart" by computing their expansion scalar - the details are in the Kinematical Description section of https://en.m.wikipedia.org/wiki/Congruence_(general_relativity).

 

[timmdeeg]

Thanks for this explaination and the link. Yes that's really helpful to improve my understandig!

 

[timmdeeg]

The Meaning of Einstein's Equation

Given a small ball of freely falling test particles initially at rest with respect to each other, the rate at which it begins to shrink is proportional to its volume times: the energy density at the center of the ball, plus the pressure in the x direction at that point, plus the pressure in the y direction, plus the pressure in the z direction.

Is it correct that the wordlines of these particles are representing a "congruence of wordlines" and that the rate at which the volume of this ball shrinks or grows is given by the expansion scalar?

If yes would it be correct to describe the "expansion" of the universe as corresponding to an increasing volume of comoving galaxies?

 

[PeterDonis]

Is it correct that the wordlines of these particles are representing a "congruence of wordlines"

Yes.

and that the rate at which the volume of this ball shrinks or grows is given by the expansion scalar?

I don't know off the top of my head if it's numerically equal, but it's related. However...

would it be correct to describe the "expansion" of the universe as corresponding to an increasing volume of comoving galaxies?

No, because there's a big difference between Baez's scenario and the universe: in Baez's scenario, you are working in the rest frame of the ball (more precisely, the rest frame of its center of mass) and the ball is very small. For the universe, there is no single "rest frame" for comoving objects and the comoving objects are not restricted to a small ball--they are spread throughout the entire universe, which is spatially infinite.

 

[phinds]

this is a coordinate dependent concept

I hear this over and over, but here's my confusion. If you have two galaxies, each X units of distance in diameter and separated by a distance of 10X, then after some amount of time they will each still be X units of distance in diameter but the distance between them will be 12X units of distance. How is this coordinate dependent? Is it somehow that their diameter is not coordinate dependent but the distance between them is?

 

[PeterDonis]

If you have two galaxies, each X units of distance in diameter and separated by a distance of 10X, then after some amount of time they will each still be X units of distance in diameter but the distance between them will be 12X units of distance. How is this coordinate dependent?

"After some amount of time" depends on the time coordinate you choose. "Separated by a distance of..." depends on the space coordinates you choose.

 

[phinds]

"Separated by a distance of..." depends on the space coordinates you choose.

OK, but how can you choose a different space coordinate for the distance across the galaxies vs the distance between the galaxies?

 

[timmdeeg]

No, because there's a big difference between Baez's scenario and the universe: in Baez's scenario, you are working in the rest frame of the ball (more precisely, the rest frame of its center of mass) and the ball is very small. For the universe, there is no single "rest frame" for comoving objects and the comoving objects are not restricted to a small ball--they are spread throughout the entire universe, which is spatially infinite.

Ah ok, so there seems no possibilty to extend Baez's ball somehow to the universe.

 

[PeterDonis]

how can you choose a different space coordinate for the distance across the galaxies vs the distance between the galaxies?

I didn't say you had to. I just said "separated by a distance of...". The point is simply that "distance" is coordinate-dependent.

 

[timmdeeg]

The point is simply that "distance" is coordinate-dependent.

While if we say "increasing distances" (standing for increasing redshifts) that's invariant, right?

 

[PeterDonis]

While if we say "increasing distances" (standing for increasing redshifts) that's invariant, right?

Increasing with respect to what?

 

[timmdeeg]

Increasing with respect to what?

increasing with respect to an arbitrary observer who measures the redshift of far away galaxies.

 

[PeterDonis]

increasing with respect to an arbitrary observer

This is too vague. With respect to what property of the observer?

 

[timmdeeg]

This is too vague. With respect to what property of the observer?

With respect to a co-moving observer.

 

[PeterDonis]

With respect to a co-moving observer.

With respect to what property of a co-moving observer? You are talking about a rate of change--"increasing". A rate of change relative to what? To put it another way, you are taking a derivative of something; with respect to what are you taking the derivative?

 

[timmdeeg]

With respect to what property of a co-moving observer? You are talking about a rate of change--"increasing". A rate of change relative to what? To put it another way, you are taking a derivative of something; with respect to what are you taking the derivative?

Hm, a rate of change relative to the proper time of the observer?

 

[PeterDonis]

a rate of change relative to the proper time of the observer?

Yes. But this will only be an invariant if it's the rate of change of an invariant. So the rate of change of the observed redshift with respect to proper time will be an invariant; but the rate of change of "distance" with respect to proper time won't be, because "distance" is not an invariant, it's coordinate-dependent.

 

[timmdeeg]

Thanks, that has been a helpful course!

 

[timmdeeg]

... but the rate of change of "distance" with respect to proper time won't be, because "distance" is not an invariant, it's coordinate-dependent.

If one talkes about the expansion of the universe without further details wouldn't then the sign of the rate of change of "distance" with respect to proper time be invariant?

 

[PeterDonis]

If one talkes about the expansion of the universe without further details wouldn't then the sign of the rate of change of "distance" with respect to proper time be invariant?

It can't be an invariant if "distance" isn't an invariant.

 

[timmdeeg]

It can't be an invariant if "distance" isn't an invariant.

So the sign of the observed rate of change of the redshift doesn't correspond to a certain sign of the rate of change of distance, that sign could still be positive or negative depending on the chosen coordinates if I understand you correctly.

 

[PeterDonis]

the sign of the observed rate of change of the redshift doesn't correspond to a certain sign of the rate of change of distance

No.

that sign could still be positive or negative depending on the chosen coordinates if I understand you correctly.

Not only due to choice of coordinates. The observed rate of change of the redshift depends on whether the expansion is accelerating or decelerating--or more precisely whether it was doing so when the light we are seeing now from a particular distant galaxy was emitted.

The observed redshift itself can be related to a "rate of change of distance", but the relationship is model dependent as well as coordinate dependent. The best way to interpret the redshift is not as telling you a speed of recession, but as telling you by what factor the universe has expanded since the light you are seeing now was emitted. But that, of course, involves the scale factor, which assumes that you have made a particular choice of coordinates (it isn't even meaningful in other coordinates).

 

[timmdeeg]

No.

Got it.

Not only due to choice of coordinates. The observed rate of change of the redshift depends on whether the expansion is accelerating or decelerating--or more precisely whether it was doing so when the light we are seeing now from a particular distant galaxy was emitted.

Ok, that I've been missing.
Thanks for your great explanation!

Quite often here in PF it comes to a point where someone explains expansion of the universe doesn't mean that space expands, it means increasing distances. This seems quite vague and with no reference to coordinate dependence. But it at least points to right direction. Would you agree to that?