Heuristic approach to the Schwarzschild geometry - Accident of deeper meaning?

  • #1
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Main Question or Discussion Point

The Schwarzschild solution can be obtained by using Newton for the weak field. However it turns out that this in fact is the exact solution.

Is this coincidence or is there more to it? Opinions?

http://arxiv.org/pdf/gr-qc/0309072v3
 

Answers and Replies

  • #2
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It is rather unfortunate that nobody wishes or can engage in a discussion on this paper.

What seems to be the case, according to this paper, is that both GR and the combination of Newtonian Gravity + Galilean Relativity give the same metric.

However there is a clear difference and the difference is the interpretation of the radius (r). While under Newton + Gallilei [itex]r = c / 2 \pi[/itex] under GR this is not the case.
 
  • #3
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It's better to think of it the other way around i.e. the Lemaitre observer in the Painleve chart has spatial slices that have the Euclidean metric and use Gallilean relativity for velocity addition. See http://arxiv.org/pdf/gr-qc/0411060v2.
 
  • #4
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What seems to be the case, according to this paper, is that both GR and the combination of Newtonian Gravity + Galilean Relativity give the same metric.
No, you mis-read Visser's paper. It is not based on Galilean relativity, as is obvious from equation (5), which is the Minkowski metric of spacetime from special relativity, and on which his heuristic derivation is based. You were misled by his comment that he uses the Galilean transformation for the time coordinate at one point in his argument, but he is by no means framing his argument within the context of Galilean relativity. Also, he says his reasoning is "quasi-Newtonian", and he says that it is definitely not a rigorous derivations, it is heuristic (i.,e., hand-waving suggestiveness). Now, it's well known that when you try to combine special relativity with Newtonian gravity, you are basically forced to arrive at general relativity. Visser's preprint is really just his personal notes on something that everyone already knows, and there are actually much better expositions of this kind of heuristic derivation of the Schwarzschild metric available. But the main point is, you have completely misunderstood what this is about. It does not in any way signify that Newtonian gravity in the context of Galilean relativity implies the Schwarzschild solution.
 
  • #5
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No, you mis-read Visser's paper. It is not based on Galilean relativity, as is obvious from equation (5), which is the Minkowski metric of spacetime from special relativity, and on which his heuristic derivation is based. You were misled by his comment that he uses the Galilean transformation for the time coordinate at one point in his argument, but he is by no means framing his argument within the context of Galilean relativity. Also, he says his reasoning is "quasi-Newtonian", and he says that it is definitely not a rigorous derivations, it is heuristic (i.,e., hand-waving suggestiveness). Now, it's well known that when you try to combine special relativity with Newtonian gravity, you are basically forced to arrive at general relativity. Visser's preprint is really just his personal notes on something that everyone already knows, and there are actually much better expositions of this kind of heuristic derivation of the Schwarzschild metric available.
From the document:
"Now this is only an approximation — we have used Newton’s gravity, Galileo’s relativity, and the notion of local inertial frames. There is no fundamental reason to believe this spacetime metric once GM/r becomes large.
...
Dropping the subscript “rigid”, the invariant interval .... is an exact solution of Einstein’s equations of general relativity, Rab = 0. It is the Schwarzschild solution in disguise.
You seem to have missed the step where Visser talks about slow speeds where SR becomes Galilean relativity.

But the main point is, you have completely misunderstood what this is about. It does not in any way signify that Newtonian gravity in the context of Galilean relativity implies the Schwarzschild solution.
I am not stating and or implying that at all.

The topic is how we explain the coincidence.
 
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  • #6
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You seem to have missed the step where Visser talks about slow speeds where SR becomes Galilean relativity.
No, you are mis-reading and mis-understanding (although, to be fair, Visser's note is poorly written and misleading). The heuristic, quasi-Newtonian, plausibility argument that he presents (all his words) is based totally on the assumption of the Minkowski metric, equation (5). Do you understand that his equation (5) is the Minkowski metric? He then uses, as an approximation at low speeds, the Galilean transformation, which of course of logically inconsistent with (5), but he argues that he can plausibly do this because at low speeds it won't introduce much error. But the point is that the basic metric he is operating on is the Minkowski metric, not the Galilean metric.
 
  • #7
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You seem to conclude, rather haphazardly, what others understand or not. Furthermore I think your tone is very arrogant.

I am merely using Visser's question for an interesting topic on this forum, nothing more nothing less.

However it looks to me like you are tilting at windmills.
 
  • #8
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From my previous message: "But the main point is, you have completely misunderstood what this is about. It does not in any way signify that Newtonian gravity in the context of Galilean relativity implies the Schwarzschild solution."

I am not stating and or implying that at all.
Excuse me, but that is exactly what you are saying. Here's the quote from your previous message: "What seems to be the case, according to this paper, is that both GR and the combination of Newtonian Gravity + Galilean Relativity give the same metric." I'm pointing out that you are mistaken, because the paper does not say this. The paper says that if you try to combine special relativity with gravity, you are pretty much forced to get general relativity, which is well known.

The topic is how we explain the coincidence.
It isn't much of a coincidence. When combining Newtonian gravity with special relativity, the only real ambiguity is whether to identify Newton's time with the proper time or the coordinate time in your postulated metric. You have a 50/50 chance of guessing right. By using the Galilean time transformations, Visser is essentially making that guess. In the framework that he established, this is really an approximation, but in a more rigorous framework is exactly correct. This isn't unusual. In fact, Einstein did the same thing in his 1915 paper describing the perihelion of mercury. He didn't have the exact Schwarzschild solution at that time, so he made a couple of approximations, but we can see in retrospect they led to the exact equation of motion under the Schwarzschild solution - even though Einstein thought it was just an approximation. The fact is that the Schwarzschild solution is so simple, and so nearly constrained by all the unambiguous conditions it must meet under any reasonable theory, that there just aren't many possibilities to choose from, so it isn't surprising to land on the exact expression.
 
  • #9
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Excuse me, but that is exactly what you are saying. Here's the quote from your previous message: "What seems to be the case, according to this paper, is that both GR and the combination of Newtonian Gravity + Galilean Relativity give the same metric."
And that is what is the case as the metric is derived by using slow speeds only, and for slow speeds SR is Galilean relativity.

When combining Newtonian gravity with special relativity, the only real ambiguity is whether to identify Newton's time with the proper time or the coordinate time in your postulated metric.
The only ambiguity? What about r and [itex]\rho[/itex], e.g. the reduced and the real radius?

The fact is that the Schwarzschild solution is so simple, and so nearly constrained by all the unambiguous conditions it must meet under any reasonable theory, that there just aren't many possibilities to choose from, so it isn't surprising to land on the exact expression.
Sounds like a good point.
 
  • #10
atyy
Science Advisor
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It isn't much of a coincidence. When combining Newtonian gravity with special relativity, the only real ambiguity is whether to identify Newton's time with the proper time or the coordinate time in your postulated metric. You have a 50/50 chance of guessing right. By using the Galilean time transformations, Visser is essentially making that guess. In the framework that he established, this is really an approximation, but in a more rigorous framework is exactly correct. This isn't unusual. In fact, Einstein did the same thing in his 1915 paper describing the perihelion of mercury. He didn't have the exact Schwarzschild solution at that time, so he made a couple of approximations, but we can see in retrospect they led to the exact equation of motion under the Schwarzschild solution - even though Einstein thought it was just an approximation. The fact is that the Schwarzschild solution is so simple, and so nearly constrained by all the unambiguous conditions it must meet under any reasonable theory, that there just aren't many possibilities to choose from, so it isn't surprising to land on the exact expression.
Hmm, reminds me of http://arxiv.org/abs/gr-qc/0611006
 
  • #11
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And that is what is the case as the metric is derived by using slow speeds only, and for slow speeds SR is Galilean relativity.
No, not at all. The metric you're referring to includes both space and time components, with a negative signature, with a scale factor of c between the space and time coordinates. This cannot be derived from Galilean relativity, which has no connection between its space and time coordinates. Visser does not derive this Minkowskian form of the metric, he simply assumes it, i.e., he begins with special relativity. Now, when transforming a set of coordinates to some other coordinates, he realizes that logically he should therefore use the Lorentz transformation, but for small velicity parameters he can use the Galilean transformation without much error. But this in no way signifies that he is working in the context of Galilean relativity. He is explicitly working in the context of the Minkowski metric of special relativity. This is why your repeated claim that Visser says Newtonain gravity plus Galilean relativity lead to the Schwarzschild metric is utterly false. You seem to realize this intermittantly, because on odd numbered posts you deny ever having said it, while on even numbered posts you re-iterate the claim.

The only ambiguity? What about r and [itex]\rho[/itex], e.g. the reduced and the real radius?
It's true that there is an ambiguity in the interpretation of the radial space coordinate, but that ambiguity doesn't enter into the derivation in the same way as do the ambiguities involving the time, partly because there is no such thing (in general) as "the real radius". For example, what is the "real" radius of a black hole? It isn't possible to integrate proper distance all the way to the central singularity. But in practice this is dealt with by simply saying exactly what we mean, i.e., the Schwarzschild parameter r equals C/2pi where C is the circumference, and the Newtonian parameter r also has this meaning. We are not presented with any clear alternative association in this case, so we naturally use this in our derivations. In contrast, we have a clear choice between coordinate time and proper time, and we have to make two inconsistent choices to arrive at the Schwarzschild metric. This correspnds to the inconsistency in Visser using the Galilean transformation in the context of Minkowski spacetime.
 
  • #12
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No, not at all.
This is getting ridiculous.

How does Visser derives his formula:

First he starts with rigid coordinates and uses a Galilean transformation:

[tex]ds^2_{rigid} = -c^2dt^2_{rigid} + || d\vec{x}_{rigid} - \vec{v}dt_{rigid}||^2[/tex]

He then expands it and finally gets:

[tex]ds^2_{rigid} = -\left[c^2 - \frac{2GM}{r}\right]dt^2_{rigid} + 2\sqrt{\frac{2GM}{r}}dr_{rigid}\;{dt_{rigid} + || d\vec{x}_{rigid}||^2[/tex]

Then he writes:
Now this is only an approximation — we have used Newton’s gravity, Galileo’s relativity, and the notion of local inertial frames. There is no fundamental reason to believe this spacetime metric once GM/r becomes large.
What part of "used ... Galileo's relativity" do you not understand?

Also perhaps you wish to demonstrate that:

[tex]d\vec{x}_{rigid} - \vec{v}dt_{rigid}[/tex]

is admissible under special relativity?
 
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  • #13
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How does Visser derives his formula: First he starts with rigid coordinates and uses a Galilean transformation...What part of "used ... Galileo's relativity" do you not understand?
Stop and look at what you wrote. What you call the "rigid coordinates" is the Minkowski metric of spacetime, which forms the basis of Visser's heuristic plausibility argument. He begins with this metric from special relativity, and then "derives" the Schwarzschild metric for a spherically symmetrical gravitational field. Along the way, at one point he makes use of a Galilean transformation, which he admits is incongruous in this context, and this is what he is referring to when he says later that he has "used" Galilean relativity. This in no way implies that his "derivation" is based on Galilean relativity. It is explicitly based on the locally Minkowskian spacetime metric of special relativity. Do you dispute this?

Also perhaps you wish to demonstrate that:

[tex]d\vec{x}_{rigid} - \vec{v}dt_{rigid}[/tex]

is admissible under special relativity?
Your request makes no sense, because what you've typed there is simply an expression, not equating someting to something else, so there is no question of "admissibility". If the question you were trying (unsuccessfully) to articulate is whether Galilean transformations between inertial coordinate system is admissible in special relativity, the answer is obviously No, which is why Visser's "use" of a Galilean transformation in his heuristic plausibility argument is incongruous, as he himself admits. But this is all beside the point, which is that his "derivation" explicitly assumes the locally Minkowski metric of special relativity. Nothing that Visser says is inconsistent with this obvious fact. I'm sure if you read his paper carefully, you will realize this.

To help you along, I suggest you follow his "heuristic construction" from the beginning. He says "The heuristic construction presented in this article arose from combining three quite different trains of thought", and the first of these is

"For an undergraduate course, I wanted to develop a reasonably clean motivation for looking at the Schwarzschild geometry suitable for students who had not seen any formal differential geometry. These students had however been exposed to Taylor and Wheeler’s “Spacetime Physics” [1], so they had seen a considerable amount of Special Relativity, including the Minkowski space invariant interval. They had also already been exposed to the notion of local inertial frames [local “free-float” frames], which notion is equivalent to introduction of the Einstein equivalence principle.... By combining these ideas I found it was possible to develop a good heuristic for the weak-field metric..."

So he assumes the student is already familiar with special relativity and the Minkowski invariant interval, and this is the basis of his derivation, as shown by equation (5). He just confused you by later making use of a Galilean transformation to approximate the Lorentz transformation, which strictly speaking makes no sense in this context. This is why Visser's note is not really pedagogically useful.
 
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