Is mass the source of spacetime?

[Ibix]

If both models are asymptotically flat, which one is the one that claimed to be more accurate than the other?

One particular researcher claims that a model including gravitomagnetic effects correctly predicts galactic rotation curves without needing dark matter where a simple Newtonian model does not. This has nothing to do with our solar system and nothing to do with asymptotic flatness - both models being proposed are at the galactic scale and asymptotically flat.

I was merely noting that you are not the only person wondering if a full GR model of the galaxy would explain away dark matter. As I said, I don't think very many prople are convinced at the moment.

 

[PeterDonis]

If both models are asymptotically flat, which one is the one that claimed to be more accurate than the other? The solar system is more accurate?

The two models in question are both models of a galaxy (a generic galaxy, not our particular galaxy), not of the solar system. One includes gravitomagnetic effects, the other doesn't. Roughly speaking, one is based on Kerr spacetime, the other on Schwarzschild spacetime. Both of those spacetimes are asymptotically flat.

 

[Angelika10]

What is the reason that the stress-energy tensor curves the space-time?

The Ricci - tensor represents a volume gain and is on the left side of the field equations, on the right side is the stress-energy tensor as a source. So, stress-energy-tensor causes a volume gain. All objects in the universe are moving in a gravitational field. Having said this, how can we assume that there is something like flat spacetime? (Of cause, there is no because of the expanding universe, but leaving aside this, we assume there would be flat space without matter, don't we?)

Leaving aside this expanding universe-things, is the asymptotically flat space time kind of a background field (background spacetime) of everything in the standardmodel?

My background in GR is (I have to admit) not so deep, only some lectures on youtube and a textbook. So as I learn it by myself there might be huge gaps in my knowledge.

 

[PeterDonis]

What is the reason that the stress-energy tensor curves the space-time?

The Einstein Field Equation.

The Ricci - tensor represents a volume gain and is on the left side of the field equations

The Ricci tensor is not on the LHS of the field equations; the Einstein tensor is. The Einstein tensor is not exactly "volume gain"; but even leaving that aside, what the LHS of the EFE represents is not just "volume gain" but "volume gain or loss". For normal matter with "attractive gravity", the EFE predicts volume loss, not gain. (More precisely, the volume of a small ball of test particles will decrease.)

This paper by Baez and Bunn gives a good basic treatment:

https://arxiv.org/abs/gr-qc/0103044

Having said this, how can we assume that there is something like flat spacetime?

We assume local flatness as an approximation that is useful for certain purposes. Nobody claims that spacetime in our actual universe is actually flat anywhere; of course it isn't, since there is matter present in the universe.

we assume there would be flat space without matter, don't we?

No. There are vacuum solutions of the EFE that are not flat.

is the asymptotically flat space time kind of a background field (background spacetime) of everything in the standardmodel?

No.

There is another heuristic argument which we haven't yet mentioned, that gives a justification for using an asymptotically flat model for an isolated system like a solar system or a galaxy. It is based on the shell theorem, which says that if we have a region of spacetime surrounded by a spherically symmetric distribution of stress-energy, the spherically symmetric distribution outside the region has no effect on the spacetime geometry inside. For the case of perfectly empty space inside the region, this means the spacetime in the empty region is flat. For the case of an isolated system inside the region, this means the spacetime in the region can be well approximated as asymptotically flat.

Of course the above premise is not exactly true in our universe either; our solar system is not surrounded by an exactly spherically symmetric distribution of matter, nor is a galaxy. But it is true to a good enough approximation to make asymptotically flat models of isolated systems useful.

 

[PeterDonis]

My background in GR is (I have to admit) not so deep, only some lectures on youtube and a textbook.

I am changing the thread level to "I" based on this.

 

[Angelika10]

I was merely noting that you are not the only person wondering if a full GR model of the galaxy would explain away dark matter. As I said, I don't think very many prople are convinced at the moment.

Ah, ok. Do one of you know if somebody has tried to just calculate a metric out of the constant velocity curves? Would it make sense? I'm just thinking that there, in the really-measured weak field limit, the real metric would show itself best. If the cause for the constant velocities is a metric and not dark matter.
It would be one which is not asymptotically flat (instead, the space-time itself would vanish in the infinity, this is highly speculative, but that's why I was asking whether "energy stress tensor is the source of spacetime", not spacetime curvature, could be true. And we are in an asymptotically flat regime only in the solar system.)

 

[PeterDonis]

Do one of you know if somebody has tried to just calculate a metric out of the constant velocity curves?

What do you mean by "constant velocity curves"?

 

[rc1]

There's a very good simplified non robust (8 step) derivation of the EFE on YouTube from Stanford from memory. Wheeler said "mass tells space how to bend and space tells mass how to move" (from memory). Basically my understanding is- Einstein asked the question "what if distortion in space time was equal to energy" and this produced the EFE.

EFE introduces some obscure mathematics such as Christoffel symbols- largely you can understand the essentials without them.

I guess the derivation of EFE is only the start of GR.

 

[rc1]

Some of the questions Angelika is asking seem to be on the edge of physics- things that physicists are struggling with- that no one knows the answer to- the question about what is the fundamental reality of "our" universe. The approach that seem to me to have the most potential is cellular automa- the question is how to build complex physical laws from simple ones that are perhaps created randomly by chance. Lee Smolin and Leonard Susskind have had interesting discussion on related things.

 

[Angelika10]

The Ricci tensor is not on the LHS of the field equations; the Einstein tensor is. The Einstein tensor is not exactly "volume gain"; but even leaving that aside, what the LHS of the EFE represents is not just "volume gain" but "volume gain or loss". For normal matter with "attractive gravity", the EFE predicts volume loss, not gain. (More precisely, the volume of a small ball of test particles will decrease.)
This paper by Baez and Bunn gives a good basic treatment:
https://arxiv.org/abs/gr-qc/0103044

Thank you for the paper! I will have a look on it! Hm... volume loss... I have to check this. Was sure until now that it's a volume gain...

...It is based on the shell theorem, which says that if we have a region of spacetime surrounded by a spherically symmetric distribution of stress-energy, the spherically symmetric distribution outside the region has no effect on the spacetime geometry inside. ...

Has the shell theorem an analogy in electromagnetic fields? If I assume a spherically symmetric distribution of current density tensor, surrounding a region - is there NO field inside?

 

[Angelika10]

What do you mean by "constant velocity curves"?

The velocity of the stars at the edges of many galaxies appears to be independent of the radius r. The reason for the dark matter assumption.

 

[PeterDonis]

Has the shell theorem an analogy in electromagnetic fields? If I assume a spherically symmetric distribution of current density tensor, surrounding a region - is there NO field inside?

The theorem in electromagnetism is that the electric field inside a hollow conductor with no charge-current density inside must be zero. This is not quite analogous to the shell theorem for gravity, although it has some similarities.

 

[rc1]

Dark Matter is an attempt to explain why Galaxies appear to have more mass than the light in the Galaxy indicates.

Dark Energy is another thing- and appears to be relate to the expansion of the universe- some explanations involve exotic matter (negative mass and energy) from memory.

 

[PeterDonis]

The velocity of the stars at the edges of many galaxies appears to be independent of the radius r. The reason for the dark matter assumption.

Ah, ok, you are talking about galaxy rotation curves. See further comments below.

If the cause for the constant velocities is a metric and not dark matter.
It would be one which is not asymptotically flat (instead, the space-time itself would vanish in the infinity, this is highly speculative, but that's why I was asking whether "energy stress tensor is the source of spacetime", not spacetime curvature, could be true. And we are in an asymptotically flat regime only in the solar system.)

None of this is correct. Most of it does not even make sense. Spacetime can't "vanish in the infinity". It makes no sense to say the cause for galaxy rotation curves is "the metric", because the metric itself depends on the distribution of stress-energy. "Dark matter" is just a way of describing the fact that the distribution of stress-energy that astronomers have to assume to account for galaxy rotation curves does not match the distribution of visible matter--there must be additional matter that is not visible. That's why the only alternatives to the dark matter hypothesis involve modifying GR itself--modifying the physical law that links the distribution of stress-energy with the motion of matter.

Finally, it is not correct that "we are in any asymptotically flat regime only in the solar system". I have already explained why in several previous posts in this thread.

 

[Angelika10]

Newton perfectly fits for the solar system, but not for whole galaxies. Of course it can be dark matter. But to gain alternative approaches, one could ask: What is the difference between the solar system and the whole galaxy? One big difference I find is that the solar system surrounds the galaxy-center, which possesses a much bigger mass then itself. The galaxy is not moving so fast around a bigger mass.
What follows from the movement of the solar system around the galaxy center?

 

[Angelika10]

In this article:
https://link.springer.com/article/10.1140/epjc/s10052-017-4695-y
they explain that introducing new fields to the EFE and introducing unknown particles is equivalent. Therefore, modifying GR is not an alternative to dark matter, its conceptually the same.

 

[PeterDonis]

Newton perfectly fits for the solar system, but not for whole galaxies.

This is not correct. Newtonian gravity with an appropriate mass distribution can fit galaxy rotation curves just fine. Newtonian gravity is not limited to point masses with an inverse square gravitational force.

What is the difference between the solar system and the whole galaxy?

The mass distribution; the solar system is well approximated by a small number of point masses interacting with an inverse square force. The galaxy as a whole is not; it is better approximated by a continuous mass distribution with a density that varies with position. (Strictly speaking, one could try to model a galaxy as a large number of interacting point masses, but this is computationally intractable because the number of point masses required would be so large--a hundred billion or more.)

The galaxy is not moving so fast around a bigger mass.

Yes, it is. Our galaxy is part of the Local Group, which in turn is part of a larger galaxy cluster, which in turn is part of a supercluster.

What follows from the movement of the solar system around the galaxy center?

Um, the fact that the galaxy is composed of a large gravitating mass?

I'm not sure what you're trying to say here.

 

[PeterDonis]

they explain that introducing new fields to the EFE and introducing unknown particles is equivalent.

That is basically correct, but it doesn't mean what you think it means. See below.

Therefore, modifying GR is not an alternative to dark matter, its conceptually the same.

Wrong. Modifying GR means modifying the EFE itself. Dark matter just means modifying the stress-energy distribution, not the EFE.

 

[vanhees71]

Yes, the EM field can be interpreted as a curvature in an internal, abstract "space". But that is not the same as saying that the EM field is curved. It's the internal abstract space that is curved (in this interpretation).

Yes, but mathematically it's the same concept, namely gauge invariance. GR can also be understood as a gauge theory. The global symmetry gauged is the Poincare group, i.e., the symmetry group of Minkowski space, and that's why in this case the connection, pseudo-metric, curvature etc. refer to the spacetime geometry and not to an "internal abstract space" (a fiber bundle).

 

[rc1]

And the Galaxy center? We're moving with 220km/s around the galactic center, in comparison to 30 km/s Earth around the sun.
Therefore, the field of the galactic center is big at our place!
But, since we're moving in free fall (with the solar system), I would suppose we do not measure any derivation from the flat space in the solar system.

Angelika10 said- "Therefore, the field of the galactic center is big at our place!"- Newton's 1st Law (of linear motion)- Speed is not dependent on force. In this case the motion is rotational motion w= v/r. Centripetal Force = mv^2/r and this is proportional (equal) to the field strength. Though the masses are the same the radii are different. The Earth is 150 M km from the Sun but much further from the Galactic Core (27,000 ly- 1 ly = 10 T km= 10x 10^12 km). So the greater the radius the smaller the Centripetal Force.
In fact if the field due to the Galactic core was greater than the Sun I suspect that the Galactic tidal forces would tear apart the Solar System.

I could be wrong... this is just a classical analysis.

 

[rc1]

You could probably argue that spacetime is approximately flat on the scale of the Solar System.

 

[Ibix]

You could probably argue that spacetime is approximately flat on the scale of the Solar System.

No you couldn't - the planets wouldn't orbit. You could argue that the contribution to the total curvature from masses outside the system is negligible on the solar system spatial scale and thousand year timescale.

 

[rc1]

No you couldn't - the planets wouldn't orbit. You could argue that the contribution to the total curvature from masses outside the system is negligible on the solar system spatial scale and thousand year timescale.

Yes sorry that's what I meant. Thanks Ibix for the correction.

Also Murray Gell- Mann has a TED Talk on why physics is beautiful at different scales because it's similar.

 

[Nugatory]

In fact if the field due to the Galactic core was greater than the Sun I suspect that the Galactic tidal forces would tear apart the Solar System.

Don’t think so... if two gravitating bodies produce the same field strength at a given point, the tidal effects are weaker for the larger and more distant one.

(No GR needed for this calculation, the Newtonian approximation is fine).

 

[Angelika10]

The Ricci tensor is not on the LHS of the field equations; the Einstein tensor is. The Einstein tensor is not exactly "volume gain"; but even leaving that aside, what the LHS of the EFE represents is not just "volume gain" but "volume gain or loss". For normal matter with "attractive gravity", the EFE predicts volume loss, not gain. (More precisely, the volume of a small ball of test particles will decrease.)

This paper by Baez and Bunn gives a good basic treatment:

https://arxiv.org/abs/gr-qc/0103044

ok, they write, that a volume of a small ball of testparticles will decrease in time. And that's the basic meaning of "gravity attracts".

My question is another: If I the mass is increased, will the volume increase or decrease because of the additional mass?
... can the metric tensor help? diag(B, -A, -r², -r²sin(theta))

 

[PeterDonis]

they write, that a volume of a small ball of testparticles will decrease in time. And that's the basic meaning of "gravity attracts".

More precisely, that the normalized "acceleration" of the volume--the second derivative of the volume with respect to time, divided by the volume--is negative. That means the volume, if it starts from "rest" (zero rate of change with time), will decrease at a rate that increases with time.

If I the mass is increased, will the volume increase or decrease because of the additional mass?

The Ricci tensor does not describe "mass". It describes density of stress-energy. For the simple case of a perfect fluid, the Ricci tensor--more precisely, the piece of it that determines the (negative) "acceleration" of the volume described above--will be ##\rho + 3 p##, where ##\rho## is the energy density of the fluid and ##p## is the pressure. So positive #\rho + 3p## means the "acceleration" of the volume is negative--"attractive" gravity. Note that this is for an observer who is inside the fluid, observing the behavior of a small ball of test particles that is also inside the fluid.

can the metric tensor help? diag(B, -A, -r², -r²sin(theta))

Where is this metric tensor coming from?

 

[Angelika10]

The Ricci tensor does not describe "mass". It describes density of stress-energy. For the simple case of a perfect fluid, the Ricci tensor--more precisely, the piece of it that determines the (negative) "acceleration" of the volume described above--will be ##\rho + 3 p##, where ##\rho## is the energy density of the fluid and ##p## is the pressure. So positive #\rho + 3p## means the "acceleration" of the volume is negative--"attractive" gravity. Note that this is for an observer who is inside the fluid, observing the behavior of a small ball of test particles that is also inside the fluid.

Hm, ok. I understand this. But the question is: if the energy-momentum-tensor is increased, what happens with the volume?

Where is this metric tensor coming from?

From the standard form of a static, spherically symetric metric.

## ds^2 = B(r)c^2dt^2 -A(r)dr^2 - r^2(d\theta^2+sin^2 \theta d\phi^2) ##

so

## (g_{\mu\nu}) = diag (B(r), -A(r), -r^2, -r^2 sin^2\theta)##

 

[PeterDonis]

if the energy-momentum-tensor is increased

What does it even mean to "increase" a tensor? A tensor is not a number.

From the standard form of a static, spherically symetric metric.

A non-vacuum spacetime (i.e., a spacetime with a nonzero Ricci tensor, which requires a nonzero stress-energy tensor) will not necessarily be either static or spherically symmetric.

 

[vanhees71]

More precisely, that the normalized "acceleration" of the volume--the second derivative of the volume with respect to time, divided by the volume--is negative. That means the volume, if it starts from "rest" (zero rate of change with time), will decrease at a rate that increases with time.The Ricci tensor does not describe "mass". It describes density of stress-energy. For the simple case of a perfect fluid, the Ricci tensor--more precisely, the piece of it that determines the (negative) "acceleration" of the volume described above--will be ##\rho + 3 p##, where ##\rho## is the energy density of the fluid and ##p## is the pressure. So positive #\rho + 3p## means the "acceleration" of the volume is negative--"attractive" gravity. Note that this is for an observer who is inside the fluid, observing the behavior of a small ball of test particles that is also inside the fluid.

But this is not the Ricci but the energy-momentum tensor, ##T_{\mu \nu}=(\epsilon+P) u^{\mu} u^{\nu}-P g^{\mu \nu}## (for an ideal fluid). The source of the gravitational field (or equivalently space-time curvature) is the energy-momentum-stress of the "matter", as described by the Einstein field equations,
$$R_{\mu \nu}-\frac{1}{2} R g_{\mu \nu} = \kappa T_{\mu \nu}.$$
Here ##R_{\mu \nu}## is the Ricci tensor, ##R## its trace, and ##\kappa=8 \pi G/c^4## with ##G## being Newton's gravitational constant. The sign on the right-hand side depends on the convention, i.e., how you contract the Riemann curvature tensor to the Ricci tensor.

An equivalent form is
$$R_{\mu \nu}=\kappa \left (T_{\mu \nu}-\frac{1}{2} T g_{\mu \nu} \right), \quad T=T_{\mu}^{\mu}.$$

 

[PeterDonis]

this is not the Ricci but the energy-momentum tensor

In the paper by Baez and Bunn that I linked to, they rearrange the field equation so that the Ricci tensor is on the LHS, in the "equivalent form" you give at the end of your post. That is because the Ricci tensor is the tensor that has the direct physical interpretation being discussed (the "acceleration" of the volume, divided by the volume, of a small ball of test particles, caused by "mass" or more precisely density of stress-energy). That is what I was referring to.

 

[catlove]

Tenors (field vectors) are associated with an entity that consitutes matter yet in vacuum there is not matter so tenor theory (guess) is speculative.

 

[PeterDonis]

Tenors (field vectors) are associated with an entity that consitutes matter

Not necessarily. Spacetime itself is described by tensors (metric tensor, Riemann tensor, etc.) even in the absence of matter.

so tenor theory (guess) is speculative.

Your guess is incorrect.

 

[catlove]

Really?

"Maxwell's electrodynamics proceeds in the same unusual way already analyzed in studying his electrostatics. Under the influence of hypotheses which remain vague and undefined in his mind, Maxwell sketches a theory which he never completes, he does not even bother to remove contradictions from it; then he starts changing this theory, he imposes on it essential modifications which he does not notify to his reader; the latter tries in vain to fix the fugitive and intangible thought of the author; just when he thinks he has got it, even the parts of the doctrine dealing with the best studied phenomena are seen to vanish. And yet this strange and disconcerting method led Maxwell to the electromagnetic theory of light!" (Duhem, 1902).

 

[PeterDonis]

Really?

Yes.

Duhem, 1902

A reference from 119 years ago, particularly from a philosopher and not a scientist, more particularly one that makes incorrect claims (Maxwell electrodynamics is perfectly consistent), and even more particularly when the issue it is talking about has nothing whatever to do with tensors, is not a good basis for discussion.