Talk:Schwarzschild metric

Content split

I've taken out a chunk of this article and extended it to give a more detailed derivation of the Schwarzschild metric in deriving the Schwarzschild solution (for those interested in such things). —The preceding unsigned comment was added by 139.133.7.37 (talk • contribs) on 16:32, 29 March 2005.

Units?

Not setting c = 1 in articles on general relativity seems completely ridiculus, for any number of reasons. I can see how, pedagogically, one might wish to do that in articles on special relativity (although I personally object). By the time someone is comfortable with reading articles on general relativity, however, they should be comfortable with the idea of natural units. In the interior of a Schwarzschild black hole the t and r coordinates become spacelike and timelike respectively. Are we now measuring time in meters and space in seconds, or do we suddenly switch the units of t at the event horizon??? -- Fropuff 1 July 2005 15:38 (UTC)

Whether you think it's "completely ridiculous" is not really the point. True, when someone is comfortable with reading GR articles, putting c=1 simplifies the equations etc. Putting c=1 or not has nothing to do with the nature of the coordinates; only the metric signature determines that. Let's agree that if a mathematical quantity is spacelike, this means that its inner product with itself is +ve; this does not necessarily mean that the quantity is measured in units of metres !!! For example, in GR, consider the four-velocity of a material particle: it's inner product with itself is always negative, but four-velocity is not measured in metres or seconds !!! Anyway, for someone who first comes across this article (and who isn't a specialist, but may have heard of black holes etc.), they're probably wondering why the units are messed up in the metric. In other articles where this metric is mentioned, I agree that putting c=1 is ok, as long as this is stated (like in some of the GR articles). Specialists have this tendency to be as elegant as possible, but they sometimes overlook the fact that not all articles are intended for them. We should remember this. --Mpatel 11:32, 17 July 2005 (UTC).

I actually came over into the Talk section here because I was rather surprised to see that the article was not using natural units. I think it's reasonable to use natural units in the article as long as we mention that is what we're doing. An explanation as to why this makes sense even beyond just simplifying the equations (i.e. treating space and time as equivalent geometric quantities, etc.) would also be appropriate. This will serve to keep the article in sync with current practice in the field while still educating and providing an introductary path for newcomers. 72.130.178.52 04:33, 5 December 2006 (UTC)

It is mentioned elsewhere (on Kerr metric page, for example) that time and space inside horizon swap. It would be nice if this will be explained a bit.

—The preceding unsigned comment was added by 195.66.192.167 (talk • contribs) on 11:04, 17 July 2005.

In 1-dimensional spacetime (one spatial coordinate) Schwarzschild metric is (in natural units, c=1)

ds^2 = -(1-rs/r)*dt^2 + dr^2/(1-rs/r)

where rs - Schwarzschild radius.

Let's use a = (1-rs/r). r belong to [0,+inf) -> a belong to (-inf,1).

ds^2 = -a*dt^2 + dr^2/a

Null geodetics (ds^2=0):

dr^2 = a^2*dt^2 (again, remember that -inf<a<1)

I've made a picture with light cones and wordlines of photons, see http://195.66.192.167/linux/blackhole_1d.gif (feel free to add it to wiwkipedia, I am new and do not ko=now how to upload it). '+' signs show 'future': regions where ds^2 > 0 (and how mathematically future is different from past? both have ds^2 > 0...).

It visualizes the following:

1) photon will never reach horizon in 'our' (distant observer's) frame of reference.

2) inside horizon directions where ds^2 > 0 are spacelike. (time and space are swapped).

Open questions:

1) I placed '+' inside horizon so that light is falling into singularity and not away from it to horizon, but this is a bit arbitrary. Is there solid reason why it is so?

2) Will photon which is somehow got inside horizon ever reach singularity in our frame of reference? I think it wouldn't, exactly like 'external' photons could not reach horizon due to time dilation. I infer time dilation from ever shrinking angle of light cone when light approaches horizon from outside or when it approaches singularity.

—The preceding unsigned comment was added by 195.66.192.167 (talk • contribs) on 11:58, 17 July 2005.

cool you just answered my question on the solution inside the black hole. but then we'd have 3 dimensons of time and 1 space!? wat does that mean? anyway i dont think its legitmate to solve it inside the horizon. quantum effects are likely to dominate (dunno about just below the horizon) —The preceding unsigned comment was added by Protecter (talk • contribs) on 11:14, 25 October 2005.

Merge from Schwarzschild black hole

Merger sounds good. ---Mpatel (talk) 18:21, August 30, 2005 (UTC)

I agree. I'll go ahead and do it.--Bcrowell 03:15, 3 September 2005 (UTC)

Isotropic coords?

Can someone add pretty language about isotropic coordinates? That is,

$\rho = \frac{1}{2} \left[r-M+\sqrt{r(r-2M)}\right]$

so that

$ds^2=\left( \frac{1-M/2\rho}{1+M/2\rho}\right)^2dt^2 - \left(1+\frac{M}{2\rho}\right)^4 (d\rho^2 + \rho^2 d\Omega^2)$

or should I just uncermoniously copy this into the article at some point? I don't think I can say anything intelligent about these coords. linas 20:19, 29 October 2005 (UTC)

Hmm, there's an article isotropic coordinates which doesn't mention this form, and this article doesn't link.linas 20:49, 29 October 2005 (UTC)

Hmm, can the above be called the "standard isotropic Schwarzschild coords" or something like that? Then the above formulas can be added to the article on isotropic coords, and this article can then be made to link to that. ?linas 21:09, 29 October 2005 (UTC)

Controversy

There is a controversy between the Schwarzschild Model for the Black Hole and the E=mc2 equation. I will be starting an article on this soon. Even better, someone help me start it. I really don't have that much time. Freddie 02:23, 20 February 2006 (UTC)

If this article contains your own thoughts on the matter, be advised that it is "original research", and will likely be deleted very quickly as a result (Wikipedia is only supposed to summarize information found elsewhere, not be a place to post new information). If you have questions about how the Schwarzschild metric and aspects of general relativity (like the mass/energy equivalence) relate to each other, a suitable place to discuss this is either on this talk page, or at Talk:General relativity). --Christopher Thomas 04:26, 20 February 2006 (UTC)

Proposed merge from Deriving the Schwarzschild solution

This merge was proposed on 3 March 2006 by User:Hillman. I've created this heading so that we can figure out if people want the merge to occur. --Christopher Thomas 20:40, 27 August 2006 (UTC)

• Oppose. I think the derivation is long enough that it's reasonable to put it in its own article to avoid clutter in this one. --Christopher Thomas 20:40, 27 August 2006 (UTC)
• Oppose. The derivation is indeed a lengthy one, which deserves its own article. Since the main article is (for the most part) qualitative, adding the derivation will not add significantly to the content of the article. --Masud 17:59, 25 September 2006 (UTC)

Year of finding

It is said in the article, that Schwarzschild found the solution in 1915. Now I am reading Landau & Lifshitz's "Classical Theory of Fields" (Polish ed., PWN, Warsaw 1976) and it is stated on p. 339 that it rather should be 1916. I am not sure if they mean year of publication, and the actual finding could take place in 1915, so I point it out in discussion, rather than edit it by myself. Paweł Laskoś-Grabowski 81.219.231.40 16:43, 8 July 2007 (UTC)

1915 is correct. Schwarzschild's letter to Einstein, dated 22 December 1915 and containing the solution, is in Einstein's archive which was not available to researchers in 1976. I posted it here. JanPB 19:06, 16 August 2007 (UTC)

Isotropic equation

Why is there an ellipsis on the last equation in the isotropic section? —Preceding unsigned comment added by 131.215.195.228 (talk) 18:45, 5 August 2010 (UTC)

As near as I can tell, this was to move the superscript for the citation link to avoid making it look like "c³". I've replaced it with whitespace instead. --Christopher Thomas (talk) 19:51, 5 August 2010 (UTC)

Original metric

The metric known (presented here) as "the Schwarzschild solution" was suggested by Einstein. But in the original Schwarzschild’s solution (see his letter - the last page in ) r was originally equal to $R = (r^3+{r_s}^3)^{1/3}$ without gravitational singularity. This my edit was reverted. Please discuss it.

Even if your reference is correct, the place where you put your edit was inappropriate. If it is referring to a different solution, then it should be in another article. If it is referring to this solution, then it should have been in a new section by itself.

However, I strongly doubt your reference since it has been established that there is a real physical singularity at the center of an spherically symmetrical uncharged black hole. See the section Schwarzschild metric#Singularities and black holes. JRSpriggs (talk) 14:13, 11 January 2011 (UTC)

So I will add a new section - "History". The original Schwarzschild solution was with $R = (r^3+{r_s}^3)^{1/3}$ instead of r in the currently named "Schwarzschild" solution. This solution has not the gravitational singularity . The current version of solution is due to David Hilbert.

Are you saying that Schwarzschild worked with R as his radial coordinate (calling it "r") instead of what we use here, r=circumference/(2π)? If so, then when R=r_s, one should have r=0. That is, anything reaching that altitude would be crushed horizontally to zero width. And you do not consider that to be a singularity? JRSpriggs (talk) 10:14, 12 January 2011 (UTC)

You can see (the last page of Schwarzschild’s letter) that r is coordinate and at r=r_s is R=1.26r_s and at r=0 is R=r_s. This is only a transformation. Nevertheless the differential form of metrics is also transformed into dR. This means that R "in the same" metrics can not reach values below r_s (horizon) for any r, but for metrics directly with r can. —Preceding unsigned comment added by 194.228.230.250 (talk) 11:57, 12 January 2011 (UTC)

I was having difficulty making sense of what you were saying. But I see from the translation of Schwarzschild's letter (to which you referred) that he uses R for that for which we use r, that is, the circumference divided by 2π. Regardless of the symbol used for the radial coordinate, the geometry and physics of the solution are unaltered. So I still fail to see how you can say that there is no physical singularity. Could you explain that in more detail please. JRSpriggs (talk) 12:33, 13 January 2011 (UTC)

If I understand correctly, the "original" radial coordinate was zero at the event horizon, so the "original" coordinate system didn't cover the interior of the black hole at all. A. di M. (talk) 14:21, 13 January 2011 (UTC)

┌────────────────────────────────────────────────────────────────────────────────────────────────────┘ Well, it appears that at that time, Einstein was still acting under the misconception that one should only use coordinate systems for which $\sqrt{- g} = 1 \,.$ This may have led him to exclude the interior of the black hole from his coordinate chart. But the singularity is there whether the coordinate chart reaches it or not. JRSpriggs (talk) 07:30, 14 January 2011 (UTC)

Can the references cited here (notes 1-4 in the article) be considered reliable sources? 2-4 are pointing to a website that's basically claiming most 'experts' on general relativity are lying. This should at least be marked as a controversial view. — HHHIPPO 20:42, 11 August 2011 (UTC)

These references do not look "profesionally" but they cite original articles (Schwarzschild, Brillouin, ...) that support them. For JRSpriggs: There is no singularity bacause there is no space (you are not able to reach it - go through the event horizon in finite time). The "original metric" stops at event horizon. There is 3D hole in spacetime (inversion of "3D ball" of spacetime from Big Bang where is no space or time outside expanding Universe). 195.113.87.138 (talk) 14:50, 15 August 2011 (UTC)

An observer outside the black hole would not see an object falling into the black hole cross the event horizon, it would appear to slow down and hang suspended. However, an observer falling into the black hole would (assuming he survived that long) see himself cross the event horizon and reach the central singularity in finite proper time. JRSpriggs (talk) 21:46, 15 August 2011 (UTC)

The article stetement: "As a result of this choice of coordinate system, the original solution did not reach all the way to the center of the black hole where the gravitational singularity lies, stopping instead at the event horizon." can not be true. The coordinate r can reach 0 (R is substitution). — Preceding unsigned comment added by 195.113.87.138 (talk) 08:38, 16 August 2011 (UTC)

The original Schwarzschild paper contained the solution in the conventional coordinates (equation 14 of Schwarzschild's paper). I've removed the nonsense that said that it didn't.(Which wasn't backed by any Reliable secondary sources anyway.)TR 13:10, 16 August 2011 (UTC)

To TimothyRias: This edit by 195.113.87.138 was an error. Your edits appeared to be premised on accepting it as correct. Therefore, I reverted your edits as well as his.

What Schwarzschild called "R" is what our formula calls "r". What he called "r" is something else. It appears to me that he began his paper by taking "r" as his radial coordinate in a spherical coordinate system. So naturally, he was assuming that 0≤r. If one puts that in terms of his "R", then it becomes α≤R which is limited to the region outside the event horizon. If I am misreading his paper, please explain how. JRSpriggs (talk) 15:25, 17 August 2011 (UTC)

The "R" in Schwarzschild's paper is the conventional polar coordinate (i.e. the one we call r in the equation in this article). In his derivation he uses a different radial coordinate which he calls r, which is defined by equation 7 of his article, and is different from the usual radial coordinate.

(Note that because the Schwarschild metric is singular at the horizon it is only valid on the patch r>r_s, anyway. Strictly speaking its is also valid on the patch with 0<r<r_s, but that patch is disconnected from the other patch so (a priori) there is no connection between the two patches.) TR 15:41, 17 August 2011 (UTC)

It sounds to me like you are agreeing with me. So please revert yourself. JRSpriggs (talk) 16:39, 17 August 2011 (UTC)

I'm not agreeing with you, in the sense that Schwarzschild simply paid no attention to the ranging a validity of his solution in his article. More importantly, I'm disagreeing with the statement that was previously in the article that "The original Schwarzschild solution used a different radial coordinate system than present formulations of the Schwarzschild metric". The final solution in Schwarzschild's paper (equation 14) uses exactly the same coordinate system as the present day formulations of that solution. The range of validity of that solution was the same is it is now (r>r_s).

The claim that the modern day version of the Schwarzschild solution is somehow different than the original one, is a typical crackpot claim. The fact that the quoted sources come from the homepage of sjcrothers (a very vocal internet crackpot that somehow thinks the entire relativity community is somehow conspiring against him) should have been a clear warning of this.TR 20:11, 17 August 2011 (UTC)

Indeed Schwarzschild starts out with r as the radius of "ordinary polar coordinates", while R is some "helper quantity". He then finds a solution of the field equations using R. To figure out the geometric meaning of his R, just integrate the metric around the equator: this gives 2*pi*R. Thus it is his R, not his r, that is the same as what we call the radius r in our article, namely the circumference divided by 2 pi.

As a side effect of his original interpretation of r as radius, Schwarzschild only considers the case r>0, that is R>R_s. He doesn't say anything about the region R<=R_s. It is however obvious that his metric is also a solution to the field equations for R<R_s, as long as the central mass is smaller than R. If we call that a valid solution depends on the exact definition of valid. There are some problems like the missing connection between the two regions and the fact that the coefficients of the metric depend explicitly on R which for R<R_s is a timelike coordinate, but that's not the issue here. — HHHIPPO 06:53, 18 August 2011 (UTC)

I don't think there is any disagreement about the fact that the coordinated that Schwarzschild called R is what is currently known as the Schwarzschild radial coordinate. What Scwarzschild called r, is indeed something different. Equation 14 of Schwarzschild's paper therefore is exactly what nowadays would be called the Schwarzschild metric. That is, there is no difference between the present day formulations of the metric and the one presented by Schwarzschild, in contradiction with what the paragraph that I removed from the article claimed.

That being said there should probably an account of the history of the interpretation of the Schwarzschild solution (an in particular its singularities) in the history section. The problem here is finding a good WP:RS that discusses this history.TR 08:40, 18 August 2011 (UTC)

I fully agree. — HHHIPPO 17:51, 18 August 2011 (UTC)

I also agree (and thus "reverted"). I think that original metric looks like :$\left(1 - \frac{\alpha}{(r^3+{\alpha}^3)^{1/3}} \right)$ and thus for r=0 (centre), R=alpha (i.e. event horizon for conventional metric) is different (naked singularity) 195.113.87.138 (talk) 08:33, 24 August 2011 (UTC)

It is great that you think that, but that is not how the metric appears in Schwarzschild's paper.TR 08:43, 24 August 2011 (UTC)

Which "Schwarzschild's paper" do you mean?! In ref. 2, page 7 (in this discussion) is different (original) metric, also in ref. 4 - the last page (Schwarzschild's letter to Einstein - in German) and also in [1] ("R", which is not able to reach zero value, does not mean "r"). It can not be same. You can not change history. 195.113.87.138 (talk) 06:10, 25 August 2011 (UTC)

Equation 14 of Schwarzschild's 1916 paper is exactly the form found in all other modern sources. Saying that its not is just plain false. Schwarzschild's interpretation of the coordinate singularity at R=α (in his notation) was indeed different than the modern interpretation (he tried to identify it with the coordinate singularity normally found in polar coordinates at the origin). That does not change the fact that it is the same mathematical solution of the Einstein field equations.TR 08:35, 25 August 2011 (UTC)

┌────────────────────────────────────────────────────────────────────────────────────────────────────┘To TimothyRias: Because of the persistence of this confusion among some readers, should we not mention the issue and explain (as the previous text did) that the real physical singularity is present at the center of the black hole regardless of whether Scharzschild intended his coordinate chart to reach that point? JRSpriggs (talk) 09:24, 25 August 2011 (UTC)

I think a lot could be gained by a proper discussion of the history of the interpretation of the Schwarzschild metric (and its singularities). In particular, such a discussion could mention that Schwarzschild tried to interpret the coordinate singularity at the horizon as the center of its coordinate system.

Of course, this discussion should be based of proper secondary sources, which I have not yet found (and do not have time to hunt for). (For the record the sources in the paragraph previously present were either primary or not reliable).TR 10:16, 25 August 2011 (UTC)

(Note that this is not actually a persistent confusion among general readers, but one among crackpots like Crothers and his readers, which happen to be very vocal about it on the internet.)TR 10:16, 25 August 2011 (UTC)

It does not matter if Crothers is vocal or not. But it is historical truth that Schwarzschild’s metric and interpretation was different (his article and letter to Einstein). 195.113.87.138 (talk) 06:26, 26 August 2011 (UTC)

To TimothyRias: I can not agree. eq. 14 contains this transformation (r is normal spherical coordinate (see above eq. 6), R is a substitution mentioned also in eq. 14 (we can not use R as r - in other case we can substitute everything and obtain what we want)). The same paper also say: "Es ist also praktisch mit identisch und Hrn. Einsteins Annäherung für die entferntesten Bedürfnisse der Praxis ausreichend." / "Therefore r is virtually identical to R and Mr. Einstein’s approximation is adequate to the strongest requirements of the practice." In other words: Schwarzschild clearly (and directly) said that there is a difference (between R a r solutions). 195.113.87.138 (talk) 09:48, 25 August 2011 (UTC)

There is no such thing as a "normal spherical coordinate" when a space has curvature. In a spherically symmetric coordinate system the radial coordinate is defined up to a reparametrization. Extra input is required to fix this coordinate. In the case of Schwarzschild's R coordinate this is the condition that the coefficient of the angular part of the metric is R^2. In the case of his r coordinate, this input is hidden in the fact that he requires his metric to have determinant 1 in his x_i coordinates. There is no question that his r and R coordinates are different. (Although the difference becomes very small as R>>α, which is what Schwarzschild's remark is about.) Schwarzschild phrases his final result in terms of the coordinate R, which is how the result is still quoted (beit with another label for the coordinate).TR 10:31, 25 August 2011 (UTC)

But you can define any substitution - for example :$Q=\frac{\alpha}{R}$ and interpret it as "inside-out coordinate" (you can not interpret it that it is real space-time) and metric coefficient become :$\left(1 - Q \right)$. But it does not mean that metric change is linearly proportional to radial coordinate (it is lin. proportional to Q) instead of inversly proportional to R. You must use correct coordinate r (this is same as for integration). "r" is from 0 to inf. (these endpoints are only correct), "R" is valid from alpha to inf., "Q" is from 1 to 0 and metric coefficient is :$\left(1 - \frac{\alpha}{R} \right)=\left(1 - \frac{\alpha}{(r^3+{\alpha}^3)^{1/3}} \right)$. Schwarzschild phrases his result in R but (directly in eq. 14) stated what does this R means (this transformation to r - hidden real radial coordinate). I think that Schwarzschild’s interpretation of this R-r difference must be mentioned. And you must find out reference where is stated that there is no effect of R-r difference (cited pages are with interpretation of R-r difference that there is "no black hole" and BH is nonsense (Brillouin’s paper)). Now how I understand it: event horizon is at R=alpha (Q=1) i.e. r=0 (together with singularity) and there is finite time (proper or not) to see r=0 (naked singularity). This is due to Schwarzschild’s correct renormalisation (without influence at r=inf. but with essential effect at r=0). 195.113.87.138 (talk) 06:14, 26 August 2011 (UTC)

It is perfectly fine to make the substitution $Q=\frac{\alpha}{R}$. Just remember that this would also mean that $dR=-\frac{\alpha}{Q^2}dQ$. Q would be a perfectly fine coordinate to work with. Although it is a bit awkward in that the asymptotically flat region now occurs as Q->0. Coordinates have no intrinsic physical meaning.TR 06:43, 26 August 2011 (UTC)

There are many sources, that there is difference (new ones [2], [3], 17, [4]) but no one to support TR. I suggest to revert TR’s undo until somebody find out reference supporting TR. JRSpriggs, do you agree? 195.113.87.138 (talk) 07:14, 26 August 2011 (UTC)

None of these sources qualify as reliable. See WP:RS.TR 09:14, 26 August 2011 (UTC)

There are also discussions like [5] and something supporting TR [6] (he extend radial coordinate r from "from 0 to inf." to "from -alpha to inf." to be able reach zero with "R"- It does not look nice. And also r=alpha at angle=0 corresponds to r=-aplha at angle=pi (i.e. twice) - but where r=alpha at angle=pi lies ... - if we accept it - this leads to infinite solutions (some of them without event horizont above singularity) and this non-linear/ambiguous theory is "useless" (you can obtain what you want/observe) - if we (mathematically) use Jacobian with singularity, we obtain (physical) singularity.). Nevertheless there must be noted (in the wiki article) that there is debate/controversy and the transformation $R = (r^3+{r_s}^3)^{1/3}$ must be shown. 195.113.87.138 (talk) 08:19, 26 August 2011 (UTC)

There is no such debate in the mainstream literature. The "criticism" comes from a few crackpots that do not understand general coordinate invariance and somehow think that some coordinates are more equal than others. Saying that there is a controversy would be a typical case of WP:UNDUE weight.TR 09:02, 26 August 2011 (UTC)

You do NOT add any argument (only "crackpots" etc.). If there is no debate (as you stated)why articles such as from Christian Corda (Electronic Journal of Theoretical Physics in 2011) are published? And how is it possible that interpretations against the mainstream opinion are also published? (100 years ago and also few years ago) Answer "yes" or "no" to following questons: Should be the R-r transformation in wiki article (as original Schwarzschild’s "notation")? Is the original solution of metric different? Is conventional Schwarzschild’s metric unique solution (for given static mass)? (Hole argument) 195.113.87.138 (talk) 11:12, 26 August 2011 (UTC)

Please read WP:UNDUE.

An to answer your last questions No. (although it could be mentioned in section discussing the history of the interpretation of the Schwarzschild metric) No. Yes (up to general coordinate transformations, and analytic continuation. I.e. any static solution of the Einstein equation is diffeomorphic to an open subset of the maximally extended Schwarzschild solution.)TR 11:30, 26 August 2011 (UTC)

This is actually a pretty good discussion of the early confusion about singularities in GR. If I have time I might summarize some of it for a few paragraphs on the history of the interpretation of the Schwarzschild metric.TR 12:01, 26 August 2011 (UTC)

To 195.113.87.138: You asked "I suggest to revert TR’s undo until somebody find out reference supporting TR. JRSpriggs, do you agree?". No, I do not agree. There would be no point in our ganging up on TimothyRias since we do not agree on what to replace his version with.

Actually, I agree with Tim except on one point — I want the article to explicitly reject your position while he wants it to ignore your position as non-notable and distracting. JRSpriggs (talk) 18:28, 28 August 2011 (UTC)

One can see that the "current" Schwarzschild’s solution (Deriving the Schwarzschild solution) is a weak field approximation and can not be used for BHs. The "original" Schwarzschild’s solution is exact solution[7]. There is no difference is the presence of singularity[8] (which is also present in Newtonian gravity and it is at the point of coordinate singularity). But there is a difference in existence of the event horizon. Without this weak field approximation there is no space ("above" singularity) from information/particles can not escape in a finite time (without quantum effects) - i.e. naked singularity. In the GRT all results depends on a "proper" choice of coordinate system (Coordinate conditions). Thus only "original" solution seems to be a correct one. There are no experiments sensitive to higher order terms in metrics and therefore there is no experimental evidence of event horizon (a "shell" of BH). 213.220.236.165 (talk) 10:42, 11 September 2011 (UTC)

To 213.220.236.165: No, you are wrong. The solution described in this article is exact. JRSpriggs (talk) 05:44, 13 September 2011 (UTC)

How solution like

$g_{44}=K\left(1 +\frac{1}{Sr}\right) \approx -c^2+\frac{2Gm}{r} = -c^2 \left(1-\frac{2Gm}{c^2 r} \right)$ can be exact? 195.113.87.138 (talk) 14:26, 15 September 2011 (UTC)

(1) WP:NOTAFORUM

(2) The weak field approximation in the formula you quote is to match the solution to the Newtonian gravitation potential (which by definition is only valid in the weak field limit). If this bothers you, you will be thrilled to know, that there are exact methods to link the schwarzschild radius to the mass. (See ADM mass or Komar mass.)TR 15:09, 15 September 2011 (UTC)

┌────────────────────────────────────────────────────────────────────────────────────────────────────┘To 195.113.87.138: The derivation prior to the line you quoted was exact. The only issue remaining at that point was to determine what quantities in classical physics correspond to the two constants in the formula for the metric. For that purpose all that is necessary is that

$\lim_{r \to +\infty} \left[ r \left( g_{44} - (-c^2 + \frac{2Gm}{r}) \right) \right] = 0 \,.$

That has just one solution, specifying values of K and S. Once those values were obtained, it was seen that the approximation sign above is actually a strict equality, that is

$K\left(1 +\frac{1}{Sr}\right) = -c^2 \left(1-\frac{2Gm}{c^2 r} \right) \,.$

However, it is not necessary to show that it is a strict equality in order for this form of the Schwarzschild metric to be an exact solution of Einstein's equations which was already known before we tried to evaluate K and S. JRSpriggs (talk) 04:48, 16 September 2011 (UTC)

How can you know that the solution with K=-c2 is exact before an evaluation of K? Why $\approx$ ("a strict equality") is still in the article Deriving the Schwarzschild solution? The "original solution" (in the standard polar coordinates [9] $(1 - \frac{r_s}{(r^3 + r_s^3)^{1/3} } ) c^2 dt^2 - \frac{r^4 (r^3 + r_s^3)^{-4/3} }{1 - \frac{r_s}{(r^3 + r_s^3)^{1/3} } } dr^2 - (r^3 + r_s^3)^{2/3} (d\theta^2 + \sin^2\theta \, d\varphi^2)$ ) is also flat for infinite r etc. (but it is different - TimothyRias agree that there are many solutions - if it is possible to obtain more "exact solutions" for one problem, is it (mathematically) exact?) 195.113.87.138 (talk) 06:07, 16 September 2011 (UTC)

To your question: Yes. The phenomenon is called gauge invariance. (or more specifically for GR general coordinate invariance.) The "different solutions" actually are just different coordinate representations of the same mathematic object (a pseudoriemannian manifold.TR 08:10, 16 September 2011 (UTC)

Ok. But mathematically correct answer/solution is: A given problem has infinitely many solutions ... one of them is "original Schwarzschild's metric" another is "conventional Schwarzschild's metric" etc. And not that (only) "conventional Schwarzschild's metric" is exact solution (and silence ...). So, why "original Schwarzschild's metric" is not mentioned in the history section? (Also note that the "conventional solution" is not diffeomorphic to "original solution" at event horizon and singularity. - due to Jacobian matrix) 195.113.87.138 (talk) 06:46, 19 September 2011 (UTC)

The final solution that Schwarzschild gives in his paper (eq 14) IS the "conventional" solution. Can you please stop trying to argue that the sky is not blue. (Also note that neither the "conventional" nor the "original" solution is mathematically defined at the event horizon, making any remark about them being diffeomorphic there or not just evidence of you lack of understanding of differential geometry and thereby by extention GR.) TR 07:58, 19 September 2011 (UTC)

If one person expressed r and the other distances involved in terms of feet and another person expressed them in terms of meters, would you say that those were different solutions? If not, then there is no reason to say that a change in the coordinate system represents a different solution. JRSpriggs (talk) 22:52, 19 September 2011 (UTC)

Please answer. Are "conventional" and "original" solution diffeomorphic (for whole space - all points - not only outside as Birkhoff's theorem (relativity)) or not? In the case yes (you say: "I.e. any static solution of the Einstein equation is diffeomorphic to an open subset of the maximally extended Schwarzschild solution."), please cite proof. In the case no, "conventional" and "original" are different (and different consequences) but "original" is not mentioned in wiki article. (And in Deriving the Schwarzschild solution is not mentioned that some gauge was selected.) Eq 14 IS NOT "conventional" solution. Imagine: You can define Q=r/2 but it does not mean that the speed of light was changed. This substitution must be taked into account (and result transformed back to "r" with "SI metre" as in eq. 14 - "R" is different). 195.113.87.138 (talk) 06:41, 20 September 2011 (UTC)

Your assertion here is just plain wrong. In eq 14 of schwarzschild's paper R is treated as a coordinate (witnessed by the dR in the expression). It is this coordinate that is now known as the Schwarzschild radial coordinate. Your assertion that the expression only has meaning if it is transformed back to the r coordinate Schwarzschild used as an intermediate step, just illustrates your lack of understanding of the role of coordinates in general relativity.TR 07:55, 20 September 2011 (UTC)

It is not the case of units (but SI second is defined for local inertial system (and metre is directly linked via c) - i.e. this unit is different in different systems (gravity/metric), but timescale like UTC is defined/specified at rotating geoid). 195.113.87.138 (talk)06:41, 20 September 2011 (UTC)