More precisely, stress-energy curves spacetime.Angelika10 said:Mass curves the spacetime.
"Curves an electric field" makes no sense; an electric field is not the kind of thing that can be curved.Angelika10 said:In electrodynamics, an electron curves an electric field
Of an electromagnetic field, In the sense that it has a charge-current density, yes.Angelika10 said:The electron is the source of an electric field
Yes, but not the one you describe in the title of this thread. The analogy is:Angelika10 said:Is there an analogy to space-time
Because it's not the kind of thing that can be curved. The concept doesn't make sense.Angelika10 said:Why can't an electric field not be curved?
No.Angelika10 said:If I imagine a constant electric field, with or without an electron, then the field is curved with the electron, isn't it?
No, I said stress-energy is the source of curvature. But there is plenty of other stress-energy besides the solar system.Angelika10 said:In the solar system, mass is the source of the curvature, as you said.
No. The galaxy is just more stress-energy, which contributes to the overall spacetime curvature in the solar system. There is no separation of "background field" and "solar system". We might choose to make a mathematical model in which we separate those things for convenience in calculation, but that's a feature of our mathematical model, not reality.Angelika10 said:the solar system experiences a background field in which the solar system is falling freely in direction to the Galaxy center.
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).vanhees71 said:The electromagnetic field is also a "curvature"
You appear to think that this "background field" is not spacetime curvature. But it is. A truly flat spacetime would have zero stress-energy everywhere--it would contain no gravitating masses whatever.Angelika10 said:the solar system experiences a background field
Yes.Angelika10 said:The Minkowski spacetime is flat everywhere, no masses which curve the spacetime (Minkowski Spacetime is the spacetime of special Relativity).
No.Angelika10 said:The solar system moves in direction to the center of the Galaxy in free fall. Therefore, within the solar system, there must be Minkowski spacetime as boundary condition, then the curvature of the sun and planets adds. Is it like that?
Ok, thanks!PeterDonis said:The galaxy is just more stress-energy, which contributes to the overall spacetime curvature in the solar system. There is no separation of "background field" and "solar system". We might choose to make a mathematical model in which we separate those things for convenience in calculation, but that's a feature of our mathematical model, not reality.
That's an approximation. You would need extremely precise measurements, or extremely long term measurements to detect the effect of other star systems' gravity on our solar system dynamics. Doesn't mean it's not there - just that it's negligible for almost every purpose. The obvious exception being predicting where our star system will be in the galaxy in the future - then other systems' gravity is important.Angelika10 said:Ok, thanks!
But, if we calculate with GR (leading to Newton's equation) in the solar system, we look only at the gravitational field of the central mass, in a spacetime which has flat boundary conditions.
Yes, and if we're doing that, we're ignoring the galaxy altogether, so it makes no sense to then say the "background field" of the galaxy is flat--there is no such "background field" in the model at all.Angelika10 said:Ok, thanks!
But, if we calculate with GR (leading to Newton's equation) in the solar system, we look only at the gravitational field of the central mass, in a spacetime which has flat boundary conditions.
Strictly speaking, the model you describe here is asymptotically flat, meaning that the flat "boundary" is at infinity. Which just emphasizes the point I made in my previous post just now, that this model does not include the galaxy, or indeed anything outside the solar system. The asymptotically flat boundary condition is certainly not any kind of claim that the "background field" of the galaxy is flat.Angelika10 said:if we calculate with GR (leading to Newton's equation) in the solar system, we look only at the gravitational field of the central mass, in a spacetime which has flat boundary conditions.
And the Galaxy center? We're moving with 220km/s around the galactic center, in comparison to 30 km/s Earth around the sun.Ibix said:That's an approximation. You would need extremely precise measurements, or extremely long term measurements to detect the effect of other star systems' gravity on our solar system dynamics. Doesn't mean it's not there - just that it's negligible for almost every purpose. The obvious exception being predicting where our star system will be in the galaxy in the future - then other systems' gravity is important.
The "field" you describe here--basically the Newtonian "force" that determines the orbital velocity--is not the same thing as spacetime curvature. See below.Angelika10 said: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!
More precisely, we do not measure any effects of the spacetime curvature due to the galaxy as a whole, locally in our solar system. That is because spacetime curvature is tidal gravity. In simplest terms, it is the difference in the Newtonian "force" from one side of the system to the other. (The full description of spacetime curvature is somewhat more complicated, but the simple description will do for this discussion.) Even though the overall Newtonian "force" (as manifested in orbital velocity) of the galaxy in the solar system is larger than that of the Sun itself, the difference in the Newtonian force due to the galaxy from one side of the solar system to the other is far too small to measure. That is why we can model the solar system locally using asymptotically flat boundary conditions without worrying about any spacetime curvature due to the galaxy. But, as I've said, such a model only describes the solar system. It does not describe the galaxy, or indeed anything outside the solar system at all.Angelika10 said: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.
The point, as Peter has said, is that it isn't the field you need to worry about. In GR terms that corresponds to first derivatives of the metric, which can be transformed away at a point. What matters is the tidal effects, the second derivatives of the metric, and how big they are across the solar system. In Newtonian terms these are smaller than the gravitational acceleration by a factor of approximately the radius of the solar system to the radius of our orbit in the galaxy - a few light hours to a few thousand light years.Angelika10 said:And the Galaxy center? We're moving with 220km/s around the galactic center, in comparison to 30 km/s Earth around the sun.
No. Curvature is not something you can transform away - it is always there. However, curvature due to sources other than the Sun is negligible on timescales below thousands of years at least - probably nearer millions - so you can nearly always model spacetime around the solar system as only curved due to masses in the system.Angelika10 said:1. Go inside the moving coordinate system (which is the solar system now). There, the solar system does not feel any curvature of spacetime because it is moving in free fall.
All of this is wrong.Angelika10 said:In my opinion/view of relativity, there are two possibilities: 1. Go inside the moving coordinate system (which is the solar system now). There, the solar system does not feel any curvature of spacetime because it is moving in free fall. Can I say, in the moving coordinate system is the minkowski spacetime?
2. looking from outside. There, the field is asymtotically flat as you say.
Assuming "we" are small enough that tidal gravity effects on our size scale are too small to feel, yes.Angelika10 said:curvature is still there, but as we're moving along it we do not "feel" it
Whether or not we can "feel" any tidal effects is an invariant, independent of any choice of coordinate system.Angelika10 said:inside the moving coordinate system?
Again, what you are describing is not Minkowski spacetime. It is not even "locally" Minkowski spacetime. It is Fermi normal coordinates centered on the space shuttle's worldline. (Here there are no corrections due to gravitating objects inside the shuttle.)Angelika10 said:If I look at the astronauts in the space shuttle, I suppose to see the minkowski space in which there are "in" at zero gravity: No gravity, no forces, no curved space time. They are in free fall.
This is not true of all curved spacetimes. It's only true of asymptotically flat ones, by definition. Such models are used because they are useful approximations.Angelika10 said:Why Minkowski spacetime is the spacetime which the curved space converges to in the infinity?
No.Angelika10 said:is there the possibility that the "dark matter" effect at the edges of the galaxy could come from using the boundary conditions of the solar system also for the whole galaxy?
More precisely, for deriving Newtonian gravity as an approximation for isolated systems the asymptotically flat model of Schwarzschild spacetime is used.Angelika10 said:for deriving Newton from General relativity the limit in infinity minkowskian spacetime is used
The galaxy as a whole is also an isolated system and can be modeled to a good approximation with an asymptotically flat model.Angelika10 said:how do we know that this limit also is true for the galaxy as a whole?
Note, though, that this is not a case of an asymptotically flat model being claimed to be less accurate than a model that is not asymptotically flat. It is a case of two asymptotically flat models, one claimed to be more accurate than the other. So asymptotic flatness itself is not the issue.Ibix said:A paper was recently posted here based on the idea that if one accounted for gravitomagnetic effects dark matter was not needed.
We don't. No real system is exactly asymptotically flat, because there is always more stuff in the universe outside of the system. A truly asymptotically flat system would be alone in the universe, with nothing outside it.Angelika10 said:How do we know that galaxies are asymptotically flat?
No. There are free-fall geodesic worldlines in any spacetime, whether it is asymptotically flat or not.Angelika10 said:is asymptotically flat the same as the free fall condition?
No. Asymptotically flat means that all the components of the Riemann tensor tend to zero as you go to infinite distance.Angelika10 said:And is asymptotically flat the same as the free fall condition?
An isolated galaxy in an otherwise empty spacetime would be expected to produce an asymptotically flat spacetime: seen from a distance it is indistinguishable from a point mass, so its spacetime must look like Schwarzschild or Kerr at great distances. You could solve the field equations explicitly (numerically) if you wanted to check.Angelika10 said:How do we know that galaxies are asymptotically flat?
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.Angelika10 said:If both models are asymptotically flat, which one is the one that claimed to be more accurate than the other?
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 said: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 Einstein Field Equation.Angelika10 said:What is the reason that the stress-energy tensor curves the space-time?
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.)Angelika10 said:The Ricci - tensor represents a volume gain and is on the left side of the field equations
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.Angelika10 said:Having said this, how can we assume that there is something like flat spacetime?
No. There are vacuum solutions of the EFE that are not flat.Angelika10 said:we assume there would be flat space without matter, don't we?
No.Angelika10 said:is the asymptotically flat space time kind of a background field (background spacetime) of everything in the standardmodel?
I am changing the thread level to "I" based on this.Angelika10 said:My background in GR is (I have to admit) not so deep, only some lectures on youtube and a textbook.
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.Ibix said: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.
What do you mean by "constant velocity curves"?Angelika10 said:Do one of you know if somebody has tried to just calculate a metric out of the constant velocity curves?
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...PeterDonis said: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
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?PeterDonis said:...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. ...
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 said:What do you mean by "constant velocity curves"?
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.Angelika10 said: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?
Ah, ok, you are talking about galaxy rotation curves. See further comments below.Angelika10 said: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.
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.Angelika10 said: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.)
