The Schwartzschild solution - why no Stress Tensor?

[Altabeh]

This is not a correct formulation. As I mentioned in a previous post, MTW computes the integral of \rho \, dV. Last time, I didn't provide a specific reference. You can find this calculation on pg 604 of "Gravitation", Misner, Thorne, Wheeler for the case of a star of uniform density. The result is that the integral of \rho \, dV[/itex] is different - larger - than the M that appears in the Schwarzschild metric.

<br /> <br /> I strongly suggest you to take a look at the pages 259-60 of Schutz&#039; book on GR. You are a little bit confused here! I guess you&#039;re supposing that we have to really care about gravitational potential energy contribution to the total mass which is not at all necessary!<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I&#039;ll mention something else that might not be too important now but could be later. The integral of rho*dV is not a frame (really coordinate) independent quantity. I&#039;ll say &quot;frame&quot; because it might make the meaning clearer - if you think of SR, a volume element depends on one&#039;s velocity, so it depends on the choice of &quot;frame&quot;. </div> </div> </blockquote><br /> Yeah because it is valid in the Newtonian limit and this is what MTW is trying to say on page 604!<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> However, because there is a meaningful notion of a &quot;static observer&quot; in the Schwarzschild case, this can be worked around. In more general cases, the integral of rho*dV might not even be well-defined in GR. </div> </div> </blockquote><br /> Would you mind giving a reference showing this ill-definedness?<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I think it bears repeating, that here isn&#039;t anyone single concept of mass in general relativity as there is in Newtonian mechanics. This surprises a lot of people. There are, however, at least three concepts in common use in GR, and probably more - though they are all very closely related. </div> </div> </blockquote><br /> Yet this does no more good than offerring 10^{10}&amp;lt;\infty+10 instead of 10^{10}&amp;lt;\infty. In all definitions of the concept of mass in GR, we are restricted! <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I&#039;m pretty sure , for instance, that three of the commonly used masses (the Komar, Bondi, and ADM masses) exist and are equal to each other for the Schwarzschild metric. However, I&#039;m not 100% positive as I have only seen the former (the Komar mass) calcuated for that case. So, in some sense, you are probably right that the parameter M has a lot of significance, but you are not right when you assume that it&#039;s the same significance as it has in Newtonian mechanics. </div> </div> </blockquote><br /> If you mean that inside a quasi-static system with a nearly flat spacetime (or s completely static one, e.g. within the interior of a Schwartzshild BH) M is not as significant as it is in Newtonian Physics because of being defined by the density-volume integral formula, you&#039;re 100% wrong! Yet, the Newtonian definition works fine due to m/r&lt;&lt;1 for almost any star known to us so it is significant! However, MTW exactly clarifies in their equations 3 and 4 on page 604 that there are relativistic contributions to m(r) that we were not introduced to in Newtonian limit! Besides, the volume of a shell of thickness dr gives its place to &quot;proper volume&quot; thus requiring us to forget about the &quot;old&quot; integrand 4\pi r^2. Nevertheless I am really comfortable with the fact that still there is one &quot;good&quot; definition of mass which is at the very least comparable, in the range of use, to most known mass-definitions in GR, e.g. Komar or Bondi mass.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Let me urge readers to *please* do a little bit of research into the topic, and not just fire off posts based on Newtonian physics, it seems to me that we&#039;ve been going around trying to squash the same missapplications of Newtonian physics again and again and again in this thread. </div> </div> </blockquote><br /> I don&#039;t think so! The Komar mass is no more significant than the Newtonian mass even in GR. We sometimes have to believe in the power of linearized field equations so why not hold on to the old assets?<br /> <br /> AB

 

[pervect]

I strongly suggest you to take a look at the pages 259-60 of Schutz' book on GR.

OK - may I suggest that you re-read that section, then, since you have Schutz?
Here's what Schutz has to say: (the emphasis is mine, however).

Thus the total mass of the star as determined by distant orbits is found
to be the integral

M = integal 4 pi r^2 rho dr

just as in Newtonian theory. This analogy is rather deceptive, however,
since the integral is over the volume element r pi r^2 dr, which is not the
element of proper volume. Proper volume in the hypersurface t = const,
is given by

(not going to type the longish equation in since you have the text, unless there's some need, but I can't imagine what it would be)

Thus M is not in any sense just the sum of all the proper energies of the fluid elements. The difference between the proper and coordinate volume
elements is where the 'gravitational potential energy' contribution to the
total mass is placed in these coordinates.

We need not look in more detail at this; it only illustrates the care one must take in applying Newtonian interpretations to relativistic equations.

You are a little bit confused here! I guess you're supposing that we have to really care about gravitational potential energy contribution to the total mass which is not at all necessary!

Meh - I'm not really interested in long flamy posts about who is more confused. But - I do thank you for supplying a reference, and taking the time to look it up, very much!

Reading Schutz, I see that Schutz is saying what MTW is saying and also what I am saying. The integral of rho * volume in the Schwarzschild metric is not the mass M in the line element, and the reason behind this is gravitational binding energy.

This is easy to perform by calculation, the only thing you need to do is to use the proper 'proper' volume element, which is given by Schutz (and MTW).

Would you mind giving a reference showing this ill-definedness?

http://arxiv.org/abs/physics/0505004 mentions it in passing:

It is known that the volume of an object viewed from distinct inertial frames
are different physical entities that are not connected each other by a Lorentz
transformation. Consequently total energy-momentum of an object in one
frame is not connected to that in another frame, i.e., energy-momentum of
an object with a finite volume is not a covariant entity.

If you mean that inside a quasi-static system with a nearly flat spacetime (or s completely static one, e.g. within the interior of a Schwartzshild BH) M is not as significant as it is in Newtonian Physics because of being defined by the density-volume integral formula, you're 100% wrong! Yet, the Newtonian definition works fine due to m/r<<1 for almost any star known to us so it is significant! However, MTW exactly clarifies in their equations 3 and 4 on page 604 that there are relativistic contributions to m(r) that we were not introduced to in Newtonian limit! Besides, the volume of a shell of thickness dr gives its place to "proper volume" thus requiring us to forget about the "old" integrand 4\pi r^2. Nevertheless I am really comfortable with the fact that still there is one "good" definition of mass which is at the very least comparable, in the range of use, to most known mass-definitions in GR, e.g. Komar or Bondi mass.

Unfortunately, I have no idea of what you are saying here, so I doubt it's what I am saying. But it seems like a bit of a digression, and I'd suggest resolving (if possible) the issue of M being different from the integral of rho * volume before moving onto some other issue.

Unfortunately, if you can't see that MTW, I, and Schutz are all saying the same thing, and you've actually read all of them carefully, this discussion may not go anywhere. I can only suggest re-reading them carefully, (especially MTW and Schutz), re-reading what you yourself wrote, and comparing them to see where the difference lies.

 

[atyy]

@Altabeh: Yes, there is a Newtonian mass equal to the mass parameter when viewed at infinity, and this is the integral over the mass density given on Schutz p259, which is not taken over elements of "proper volume". No problems (except maybe terminology) when the vacuum Schwarzschild solution is matched to an interior solution with mass (defined as fields with localizable stress-momentim-energy). I'm curious what your preferred terminology is for the maximally extended vacuum Schwarzschild solution. Would you say there is a mass in its centre (along the lines of the OP)?

 

[Altabeh]

OK - may I suggest that you re-read that section, then, since you have Schutz?
Here's what Schutz has to say: (the emphasis is mine, however).

Meh - I'm not really interested in long flamy posts about who is more confused. But - I do thank you for supplying a reference, and taking the time to look it up, very much!.

That is not so much of pain for me to quote a paragraph of book that I deal with every day! That is you who have brought up the idea of "what is better" and "when who is right" so do not digress the discussion through playing with words! I exactly said that

I strongly suggest you to take a look at the pages 259-60 of Schutz' book on GR. You are a little bit confused here! I guess you're supposing that we have to really care about gravitational potential energy contribution to the total mass which is not at all necessary!

And you cut a quarter of it, I guess!

eading Schutz, I see that Schutz is saying what MTW is saying and also what I am saying. The integral of rho * volume in the Schwarzschild metric is not the mass M in the line element, and the reason behind this is gravitational binding energy

And I'm not saying anything else, am I?

This is easy to perform by calculation, the only thing you need to do is to use the proper 'proper' volume element, which is given by Schutz (and MTW).

Thank you for showing me the right way to do it!

http://arxiv.org/abs/physics/0505004 mentions it in passing:

And again thanks for the effort and time you've put into finding this on the Internet!

Unfortunately, I have no idea of what you are saying here, so I doubt it's what I am saying. But it seems like a bit of a digression, and I'd suggest resolving (if possible) the issue of M being different from the integral of rho * volume before moving onto some other issue.

Since you don't, then I don't see any reason to keep this going! I'm not interested in long flamy posts either!

Unfortunately, if you can't see that MTW, I, and Schutz are all saying the same thing, and you've actually read all of them carefully, this discussion may not go anywhere. I can only suggest re-reading them carefully, (especially MTW and Schutz), re-reading what you yourself wrote, and comparing them to see where the difference lies.

Did you see something in my post that made you deviate the whole thing from the worthiness of Newtonian mass in GR as in classical mechanics to a problem of "who can't see what"!?

AB

 

[Altabeh]

I've been working my way through GR using mainly D'Inverno's Introducing Einstein's General Relativity, but with MTW as well and a couple of other books. I began to get a sense of what was going to happen and got a big surprise when I reached the Schwartzschild solution. The process goes something like this - we solve the equations for a spherically symmetric vacuum and lo and behold, the solution tells us that there is a mass at the middle! But it is a mass which doesn't generate a stress tensor. I've looked through all my books and the only book that even mentions this apparent illogicality is Penrose's Road to Reality, and he just makes reference to another book by Pais. Why does no one want to talk about this? Can anyone explain?

Well to me that is not a surprise why exterior Schwarzschild solution has a mass involved with its metric! Here gravitational mass is the reason behind the curvature of spacetime around the gravitating body! And of course where you're at there is no energy nor is any sign of mass so stress-momentum tensor is expected to vanish! Yet, your spacetime is geometrically under the effect of the central mass there so one is looking for a solution that mass is not present where calculations are made, but rather the geometrical effect of mass is felt! Dealing with a zero T_{ab} does not always inspire that the spacetime is Ricci-flat! Sometimes as in this case, it also means that

R_{ab}=\frac{1}{2}g_{ab}R and therefore the spacetime under discussion is not Ricci-flat!

I hope we are clear!

AB

 

[Altabeh]

I'm curious what your preferred terminology is for the maximally extended vacuum Schwarzschild solution. Would you say there is a mass in its centre (along the lines of the OP)?

Well, why not use that terminology you gave? It is said that the mass center of Schwartzschild spacetime is at the origin! Is there any problem with that terminology, atyy?

AB

 

[Passionflower]

Thanks to Birkhoff we know that we can 'remove' the mass from the solution. So even if there is not a black hole but an existing spherical mass, the spacetime curvature outside this mass is identical to the case of a black hole. Note that this is distinctly different from a Kerr solution, which is only valid in case there is a black hole. If there is no black hole and a rotating spherical mass we cannot use the Kerr solution for the spacetime outside this mass.

 

[atyy]

Well, why not use that terminology you gave? It is said that the mass center of Schwartzschild spacetime is at the origin! Is there any problem with that terminology, atyy?

AB

Well, I go back and forth on this.

Option A: If we accept the mass at the centre, then we accept that the curvature singularity is physical, but there is no problem with matching to observations since there is an event horizon for this solution, cosmic censorship holds and there is no naked singularity.

Option B: OTOH, there is the more accepted point of view that the singularities of GR are unphysical, and imply a breakdown of the theory. In this view, it is usually said that we expect quantum gravity to replace GR at high curvatures, and the singularities will be "resolved" in such a theory.

Of course, if option A is acceptable, that doesn't mean that GR will not be replaced by some theory of quantum gravity, only that GR does not indicate its own failure at high curvatures.

 

[Altabeh]

Well, I go back and forth on this.

Option A: If we accept the mass at the centre, then we accept that the curvature singularity is physical, but there is no problem with matching to observations since there is an event horizon for this solution, cosmic censorship holds and there is no naked singularity.

Option B: OTOH, there is the more accepted point of view that the singularities of GR are unphysical, and imply a breakdown of the theory. In this view, it is usually said that we expect quantum gravity to replace GR at high curvatures, and the singularities will be "resolved" in such a theory.

Of course, if option A is acceptable, that doesn't mean that GR will not be replaced by some theory of quantum gravity, only that GR does not indicate its own failure at high curvatures.

Well I think the option one is widely accepted but there is a question then arising from the existence of a physical singularity at the center: Do you think the appearence of such singularity is a failure for GR?

I recently hit this http://www.holoscience.com/news/img/DPS%20talk.pdf" on the Internet which claims based on the falsifying Hilbert's version of Schwarzschild solution, our knowledge of Schwarzschild metric has grown up in the wrong way and the definition of "r" in this spacetime "has never been rightly identified by the physicists"! I did not take my time to read the whole paper but you might do so in order to find out if the option A lacks the defect you mentioned!

AB

 

[atyy]

Well I think the option one is widely accepted but there is a question then arising from the existence of a physical singularity at the center: Do you think the appearence of such singularity is a failure for GR?

I recently hit this http://www.holoscience.com/news/img/DPS%20talk.pdf" on the Internet which claims based on the falsifying Hilbert's version of Schwarzschild solution, our knowledge of Schwarzschild metric has grown up in the wrong way and the definition of "r" in this spacetime "has never been rightly identified by the physicists"! I did not take my time to read the whole paper but you might do so in order to find out if the option A lacks the defect you mentioned!

AB

I am quite happy (intellectually) to live with a singularity in the universe. I don't see how the singularities are mathematical inconsistencies, nor do I see any conflict with observation (in principle).

The paper you mentioned is famous on PF - it's complete nonsense.

 

[pervect]

That is not so much of pain for me to quote a paragraph of book that I deal with every day! That is you who have brought up the idea of "what is better" and "when who is right" so do not digress the discussion through playing with words! I exactly said that

You posted something that was factually incorrect. Now , it sems I'm on the defensive (or supposed to be on the defensive) for daring to mention it. Meh.

And you cut a quarter of it, I guess!

Oh - the utter horror, of trimming a reference to a book that you "deal with every day." But you certainly are good at dragging up irrelevancies that serve to derail any serious discussion of the original issue at hand, i.e. the difference between the mass M in the Schwarzschild line element and integrating rho * volume, which yield two different numbers.

To recap the discussion to date.

1) I mentioned, briefly, that the two numbers were different, and provided a textbook reference that showed the difference.

2) This gets ignored, and you suggest that "I'm confused" (which is rather rude), and you provide a reference of your own. Having access to said reference, I read it, and it doesn't support your position, and agrees with my reference.

3) I point this out, and go to the trouble of actually quoting a large section of the reference, bolding the important points. You respond by saying "you deal with the book everyday", and seem to suggest that quoting references is an unseemly way to settle an argument. I suppose it's much better to settle arguments by saying "I deal with the book everyday", an appeal to (non-existent, but presumably desired) personal authority.

Sorry - it's pretty clear to me that you aren't listening, and that you like to flame and debate rather than to discusss the actual physics involved.

 

[atyy]

@pervect: Isn't Altabeh's equations exactly the same as 10.41 on p259 of Schutz? ie. it isn't over elements of "proper volume".

 

[pervect]

@pervect: Isn't Altabeh's equations exactly the same as 10.41 on p259 of Schutz? ie. it isn't over elements of "proper volume".

While one does integrate 4 pi r^2 dr to get the mass (10.41 of Schutz), the resemblence of 10.41 to a volume element is only formal. 10.41 is *not* equivalent to integrating the volume times the density. It's an easy mistake to make if one is unwary - Schutz warns specifically against it , and I was essentially doing the same.

The proper volume element is (4 pi r^2) (dr / sqrt(1-2M/r)) for the constant density star case and

<br /> 4 \pi r^2 e^{\Lambda} dr <br />

in the more general case where the interior metric is

<br /> (e^{\Phi} dt)^2 + (e^{\Lambda} dr)^2 + (r d \theta)^2 + (r \sin \theta d\phi)^2<br />

 

[TerryW]

I understand TerryW's concerns. Sometimes the whole thing seems like magic. But that's the price we sometimes pay for elegant solutions. They are so amazing and powerful that they seem impossibly simple and even somewhat magical.

Well my post does seem to have generated a lot of interest and debate. For the moment, I'm going to carry on working with the 'accepted' paradigm but to me, this elegant solution has a big wart at the end of its nose!

Regards to all

 

[Altabeh]

You posted something that was factually incorrect. Now , it sems I'm on the defensive (or supposed to be on the defensive) for daring to mention it. Meh.

Ugh, what factually is wrong is that your eyes must be rolled over all the sentences people write here! I don't have time to help you out with that problem! Sorry!

Oh - the utter horror, of trimming a reference to a book that you "deal with every day." But you certainly are good at dragging up irrelevancies that serve to derail any serious discussion of the original issue at hand, i.e. the difference between the mass M in the Schwarzschild line element and integrating rho * volume, which yield two different numbers.

Another nonsense you're making here is that you clearly ignore the fact that still Schutz has not changed his mind on his equation 10.41. I'm still insisting that you're hugely confused because you said Newtonian mass is not as important as it is in classical mechanics! All I said is that you better look at Schutz' notes to understand we do not really need any relativistic contribution to the mass term because certainly m/r<<1 for almost all known stars\planets and stuff! It seems to me that you're like the guy who doesn't neglect the second order terms and so on in linearizing field equations because he "thinks" such terms would contribute to the results enormously. With all such nonsense claims that you've made about the importance of Newtonian mass in GR, nothing has changed the minds to stop thinking about it to date! You better understand that in almost every critical situation in GR, as in the case where we want to introduce "conservation laws", the use of Newtonian definition of mass is unbelievably excessive! If you still disbelieve it , I can quote from the "books" I've read until now! I'm not of non-scientific type!

To recap the discussion to date.

1) I mentioned, briefly, that the two numbers were different, and provided a textbook reference that showed the difference.

And if there is any confusion about this is from your side!

2) This gets ignored, and you suggest that "I'm confused" (which is rather rude), and you provide a reference of your own. Having access to said reference, I read it, and it doesn't support your position, and agrees with my reference.

Yet you're confused which this quote from atyy's post makes me strongly confortable with this credo:

@pervect: Isn't Altabeh's equations exactly the same as 10.41 on p259 of Schutz? ie. it isn't over elements of "proper volume".

3) I point this out, and go to the trouble of actually quoting a large section of the reference, bolding the important points. You respond by saying "you deal with the book everyday", and seem to suggest that quoting references is an unseemly way to settle an argument. I suppose it's much better to settle arguments by saying "I deal with the book everyday", an appeal to (non-existent, but presumably desired) personal authority.

Irrelevant! This is not psychology class and you're not the one who has authority over the others to see things from that angle!

Sorry - it's pretty clear to me that you aren't listening, and that you like to flame and debate rather than to discusss the actual physics involved.

Looking at my posts, everybody can see if I'm talking about actual physics! So let people judge!

AB

 

[Altabeh]

While one does integrate 4 pi r^2 dr to get the mass (10.41 of Schutz), the resemblence of 10.41 to a volume element is only formal. 10.41 is *not* equivalent to integrating the volume times the density.

What? Either I can't understand this or the problem is you can't understand that in the case of a spherically symmetric material distribution,

\rho=\rho(r)

with r being the distance from the center of symmetry, we clearly have

M=\int \rho d^3x.

If you're against this argument, then you're not even wrong! I can assert this by quoting another book I've read! But since you believe it is not right, then I don't!

It's an easy mistake to make if one is unwary - Schutz warns specifically against it , and I was essentially doing the same.

In general relativity, it is like a very easy essay: In case you're in a nearly flat spacetime, all contributions from gravitational energies are neglected and thus the definition gets narrowed down to

M=E/c^2=c^{-2}\int_V T_{00}dV.

And this may sound deceptive, but it is always written in that way!

The proper volume element is (4 pi r^2) (dr / sqrt(1-2M/r)) for the constant density star case and

<br /> 4 \pi r^2 e^{\Lambda} dr <br />

in the more general case where the interior metric is

<br /> (e^{\Phi} dt)^2 + (e^{\Lambda} dr)^2 + (r d \theta)^2 + (r \sin \theta d\phi)^2<br />

Nice try! We are all aware of that!

AB

 

[pervect]

@pervect: Isn't Altabeh's equations exactly the same as 10.41 on p259 of Schutz? ie. it isn't over elements of "proper volume".

Oh, the other thing I should have added is that while the initial error in confusing the volume elements may seem small and unimportant, the subsequent error in ignoring the contribution of gravitational binding energy to mass results in really, really bad self-consistency problems.

I'd be tempted to get more into how bad this is, but it's hard to find textbooks that discuss it (they focus on doing things right in the first place, not disucssing the consequences of incorrect assumptions), and the discussion isn't proceeding well for a less formal approach.

 

[atyy]

Oh, the other thing I should have added is that while the initial error in confusing the volume elements may seem small and unimportant, the subsequent error in ignoring the contribution of gravitational binding energy to mass results in really, really bad self-consistency problems.

I'd be tempted to get more into how bad this is, but it's hard to find textbooks that discuss it (they focus on doing things right in the first place, not disucssing the consequences of incorrect assumptions), and the discussion isn't proceeding well for a less formal approach.

I do understand the difference between the two integrals. But isn't Altabeh's equation the exactly correct one, since he is not integrating over proper volume elements (well, technically it's hard to tell, since the Schwarzschild solution is usually given in coordinates that don't have x,y,z, whereas his equation has dxdydz, so one would need to know how x,y,z in his equation relate to the usual coordinates).

 

[pervect]

I do understand the difference between the two integrals. But isn't Altabeh's equation the exactly correct one, since he is not integrating over proper volume elements (well, technically it's hard to tell, since the Schwarzschild solution is usually given in coordinates that don't have x,y,z, whereas his equation has dxdydz, so one would need to know how x,y,z in his equation relate to the usual coordinates).

How does "ambiguous and invites confusion" sound (rather than exactly corrrect)? Schutz's 10.41

\int 4 \, \pi r^2 \, \rho \, dr

would be "exactly correct", and would be an adequate definition of mass for a static, spherically symmetric space-time, though unfortunately it does not generalize, and it's also written in a coordinate dependent form.

One of the OTHER issues with writing d^3x, aside from the fact that it's at best unclear, is that it invites the reader to believe that the formula would apply in cases other than a static, spherically symmetric space-time, which is not the case.

The formula is coordinate dependent. The notation d^3x does not match the coordinates used, which must be the Schwarzschild coordinates for the formula to work.

 

[atyy]

How does "ambiguous and invites confusion" sound (rather than exactly corrrect)? Schutz's 10.41

\int 4 \, \pi r^2 \, \rho \, dr

would be "exactly correct", and would be an adequate definition of mass for a static, spherically symmetric space-time, though unfortunately it does not generalize, and it's also written in a coordinate dependent form.

One of the OTHER issues with writing d^3x, aside from the fact that it's at best unclear, is that it invites the reader to believe that the formula would apply in cases other than a static, spherically symmetric space-time, which is not the case.

The formula is coordinate dependent. The notation d^3x does not match the coordinates used, which must be the Schwarzschild coordinates for the formula to work.

Sure - it's just that Altabeh was commenting on one of my posts, and (not initially) after rereading his post I gave Altabeh the benefit of the doubt from my experience with his usually reliable posts, and his readiness to correct any of his technical errors (almost all disagreements are usually purely about terminology).

 

[Altabeh]

I'd be tempted to get more into how bad this is, but it's hard to find textbooks that discuss it (they focus on doing things right in the first place, not disucssing the consequences of incorrect assumptions), and the discussion isn't proceeding well for a less formal approach.

Textbooks do not go into such unipmortant things because it is of no concern in the models we have discussed to date about the structure of spacetime in a nearly flat spacetime! If you claim you got something interesting and instructive, then I'm ready to discuss it! But on a personal point, there you won't come up with anything appealing if you want to take the binding energy into account because the correction terms appearing in the equation are very tiny compared to terms up to the linear order! In the same book, on page 158, when Schutz proves a theorem on local flatness, you can see that the non-vanishing of terms like g_{ab,cd}, g_{ab,cde} in the geodesic coordinates are definitely not guaranteed so they appear in the Taylor expansion of g&#039;_{ab} meaning that in the neighbourhood of a particular point, say, P at which a coordinate basis is chosen so that the metric is flat, metric may not really be "nearly" flat according to the fact that there is an infinite sum over all terms of higher order in the metric derivatives! But this is simply negligible because infinitesimally small as an "assumption" means x^a-x^a_p are supposed to be really small that all that sum is inconsiderable!

Such assumptions as the one mentioned above are generally true even if your spacetime is stupidly curved and the the contribution of metric-derivative factors go over the top compared to the factors involving multiples of coordinate differences! But no one has ever claimed the theorem of local flatness wouldn't work somewhere around the intrinsic singularity of Schwartzschild metric!

I'm always ready to hear new ideas and if you think this is something never been worked out, it is time to shine and let us know if it deservers to be published!

AB

 

[TerryW]

Having now got as far as Chapter 18 and the Reisner Nordstrom solution, it is interesting to see that you put in the Maxwell Tensor, work through to a solution and lo, the solution contains a term which we can identify as the charge sitting at the origin. It also contains the mass sitting at the origin, but this appears without having started out with a Mass/Energy Tensor!

I'm kind of resigned to a train of thought that goes along the line that the general solution worked out from the canonical form of the metric produces a general solution which says that a space will be flat if there is no mass at the origin and curved if there is, but I can't say I'm all that happy!