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.

Do photons or spacelike separations exist in a Schwarzschild spacetime?

A metric of a Lorentzian 3+1 spacetime should allow positive, null and negative spacetime paths. The present version presents a coordinate expression for the metric using dtau^2 where tau is defined as proper time. This appears to exclude photons or any other particles following null geodesics from existing in the spacetime, and, ironically, gives a metric (informally, a method of determing length) which cannot measure any spacelike separations! The present version of our article implicitly claims that there is no space in a Schwarzschild spacetime.

I corrected this a year ago, but in this edit the following day, someone reverted my edits. The edit summary stated:

• "erroneous expression for tau;"
• "actual particle paths are time-like;"
• "let us not introduce an unnecessary integral, nor an unnecessary and overloaded variable for the path"

Respectively:

• I don't see what is wrong with the expression I gave for tau (maybe some absolute value bars could be added);
• a metric is needed for more than just timelike world lines - e.g. how can the 3-Ricci spatial curvature be defined if the metric is only for timelike worldlines?
• (two points)
  • the presentation could be made correct without the integral, but someone seemed to want proper time to be prominent here, so some other method should be chosen to describe what it is in terms of the metric;
  • another variable than "s" could be used, "s" just happens to be a common choice; "tau" in the absence of defining it as proper time is misleading (though not strictly wrong) because it makes the reader think of worldlines only; "tau" together with a definition constraining it to be proper time is wrong because it excludes null and spacelike intervals.

Boud (talk) 22:13, 18 July 2014 (UTC) Modified: added "timelike" twice - see below. Boud (talk) 21:37, 20 July 2014 (UTC)

The metric is expressed as a line element. Depending on the signature convention used this is either an infinitesimal (squared) of proper time or of proper length. Both signatures are common, this article uses the signature convention where the line element is equal dτ². I don't see how this precludes the existence of null of spacelike line segments. These are simply line segments along which dτ² is zero or negative respectively.TR 11:52, 19 July 2014 (UTC)

I have no problem with choosing a convention in which timelike intervals have positive dτ² and spacelike intervals have negative dτ². The problem is the first definition below the line element:

"where $\tau$ is the proper time (time measured by a clock moving along the same world line with the test particle) ..."

Photons don't have worldlines. Tachyonic (spacelike) paths aren't worldlines - does talking about a "clock moving along a spacelike path" make sense? Boud (talk) 22:48, 19 July 2014 (UTC)

There are certainly changes of wording that could make this correct instead of wrong. E.g. something like "where for dtau^2 positive, \tau is the proper time ..., for dtau^2 negative, ${\sqrt {-d\tau ^{2}}}$ is a spacelike interval, and zero dtau^2 applies to null intervals." Boud (talk) 22:53, 19 July 2014 (UTC)

"Photons don't have worldlines." that is simply false. "Tachyonic (spacelike) paths aren't worldlines" slightly more discutabel, but they can still be considered the worldines of tachyonic particles.TR 18:30, 20 July 2014 (UTC)

• Misner, Thorne & Wheeler p388, "Exercise 16.2 WORLD LINES OF PHOTONS". So MTW use "world lines" to apply to photons at least once.
• I haven't read any tachyonic literature, so I can't claim that "worldlines" or "world lines" is not considered an acceptable term for their spacetime paths. So I withdraw both these particular statements. These imply editorial work needed at World_line#World_lines_in_general_relativity ("Also, in general relativity, world lines are timelike curves in spacetime,"), but that's an issue there, not here. (wikt:world line gives no restriction on sign, but restricts the definition to elementary particles).

You don't seem to be claiming that "proper time" is meaningful for a photon's path or for a spacelike path. So:

• Any objections to something along the lines of: "where for dtau^2 positive, \tau is the proper time ...; for dtau^2 negative, ${\sqrt {-d\tau ^{2}}}$ gives spacelike intervals; and zero dtau^2 gives null intervals." ? Boud (talk) 21:37, 20 July 2014 (UTC)

There were no objections in about 3 weeks, so I made an edit along these lines to correct the error. User:TimothyRias reverted my edit with the edit summary "unnecessary". I don't understand why it is "unnecessary" to correct an obvious error. tau cannot be "proper time" when the interval is null or negative. Probably other people should participate in this discussion, since we don't seem to have made much progress. Boud (talk) 20:58, 13 August 2014 (UTC)

To Boud: I agree with TimothyRias (who reverted you) that your change was unnecessary. In fact, you introduced some error in your change since you left out the speed of light squared and the minus sign when referring to distance.

To describe what ${ds}^{2}$ means would be difficult and distracting. You would have to talk about a free-falling unstressed and unrotating measuring rod and somehow describe what it means for the infinitesimal interval ds to be orthogonal to the motion of the rod. JRSpriggs (talk) 05:15, 14 August 2014 (UTC)

JRSpriggs, you have ignored my main point. We cannot leave the erroneous statement in place. Please read the title of this section: "Do photons or spacelike separations exist in a Schwarzschild spacetime?" The definition of $\tau$ in this version of the article is still invalid for null and spacelike separations - two years since I first raised this point. Do you really claim that photons and spacelike separations are absent from a Schwarzschild spacetime?

Since you are worried about c^2 in non-natural units and the minus sign for an interpretation of "corresponds" to mean "is identical", I will only add the minimal correction to remove the false statement. Please see proper time. Here are some quotes (my emphasis) to help:

• "In relativity, proper time along a timelike world line is defined as the time as measured by a clock following that line."
• "Proper time can only be defined for timelike paths through spacetime which allow for the construction of an accompanying set of physical rulers and clocks. The same formalism for spacelike paths leads to a measurement of proper distance rather than proper time."
• "For lightlike paths, there exists no concept of proper time and it is undefined as the spacetime interval is identically zero."

We can omit description of null and spacelike intervals, but we cannot state that these are absent from the spacetime described by the Schwarzschild metric.

Boud (talk) 22:24, 30 October 2016 (UTC)

New section on "Curvatures"

Absolutelypuremilk (talk · contribs) just added a new section Schwarzschild metric#Curvatures. The formulas appear suspicious to me, so I added a "citation needed" template. For example, it says

$R_{1010}=-R_{2020}=-R_{3030}={\frac {2GM(2GM-r)}{r^{4}}}$

Also no value is given for $R_{0000}$, which means that it is zero. If that were true, then using the facts that the Ricci tensor is zero and that the metric is diagonal we get:

$0=R_{00}=g^{00}R_{0000}+g^{11}R_{1010}+g^{22}R_{2020}+g^{33}R_{3030}=0+{\frac {2GM-r}{r}}{\frac {2GM(2GM-r)}{r^{4}}}+{\frac {-1}{r^{2}}}{\frac {-2GM(2GM-r)}{r^{4}}}+{\frac {-1}{r^{2}\sin ^{2}\theta }}{\frac {-2GM(2GM-r)}{r^{4}}}$

which is obviously impossible (except for special values of r and θ). JRSpriggs (talk) 04:30, 25 October 2016 (UTC)

The values given were indeed wrong. More specifically they were for the form of the Riemann curvature with one index raised. (And then still there were factors of 2 missing). I've corrected them. We still need a source. "TR's mathematica notebook" is not going to cut it. Unfortunately, the half-dozen text books that I consulted all fail to give the Riemann curvature in an explicit form. (Typically, they will give the Riemann tensor for some spherically symmetric Ansatz with arbitrary functions. However they never bother revisiting the Riemann curvature once they have solve the Einstein equation for these functions.)TR 08:30, 25 October 2016 (UTC)

I indeed calculated them directly from Mathematica and missed that the first index was raised (and the factors of two). I will try and look for a source now. Absolutelypuremilk (talk) 09:45, 25 October 2016 (UTC)

This is straightforward to check in Maxima (software), which is verifiable by the community (it has 34 years of community verification), in contrast to Mathematica:

load(ctensor);
init_ctensor();
csetup();
4;
y;
[t,r,theta,phi];
1;
1;
(1-rs/r)*c^2 ;   /* t */
-1/(1-rs/r) ;  /* r */
-r^2 ;    /* theta */
-r^2 * (sin(theta))^2;  /* phi */
n;
y;
riemann(true);
riem[1,2,1,2];

Maxima/ctensor gives the opposite sign convention for the Riemann tensor, but I left this unchanged; I only inserted the $c^{2}$ for consistency with the expression for the metric. The final riem[1,2,1,2] is to help persuade maxima to simplify the expression.

Boud (talk) 23:08, 30 October 2016 (UTC)

To Boud: Thank you for correcting the spelling and adding the needed factor of c². JRSpriggs (talk) 04:23, 1 November 2016 (UTC)