Doubt about gravitational waves

[TrickyDicky]

The usual derivation of the wave form equations from the GR field equations is done in the weak field, linearized approximation of the GR theory. In this limit, that ignores non-linear contributions and that gives accurate results when used to predict solutions for problems in the Newtonian limit (classical tests of relativity, more recently rates of orbital decay of binary system pulsar-see Hulse-Taylor pulsar..) the background space is the static flat Minkowski spacetime.

So this linearised EFE when used in the theory of gravitational radiation are mixing a static spacetime that by the Birkhoff's theorem doesn't allow gravitational radiation to exist(this is related to the specific features of static spacetimes that I won't go into in this post) and the interpretation that the equations similarity to wave equations and more specifically to the tranverse EM waves of Maxwell theory (see GEM equations, etc) must mean the existence of gravitational type of waves.

I see here, that at least theoretically, something doesn't fit completely, but maybe it's just my impression, I just would like to understand gravitational waves within the conceptual framework of GR. For instance, if the linearized EFE look like EM wave equations why not consider them EM radiation to begin with?
Moreover, if the background spacetime where the linearized equations are similar to wave equations forbids gravitational type of waves, is this not a sign that maybe they are not gravitational waves, but plain EM waves radiated from the mass quadrupole (which would justify the observed orbital decay in binary systems like the Hulse-Taylor pulsar?

Of course, I'm not proposing any alternative, just would like to know how is this addressed from the point of view of mainstream GR theory, or how any of my perhaps naive premises are not correctly applied to this subject (wich surely are not).

 

[Mentz114]

I found the derivation of GWs from the linearized theory somewhat unconviving, but there are exact (vacuum) solutions of the field equations that look like plane-fronted waves with parallel rays (pp-waves). These solutions have Riemann curvature associated with the waves.

What I've never seen is a solution for, say, an oscillating mass that radiates gravitationally. I'm not sure if such a solution can exist because the EMT of the source would not be conservative, whereas the Einstein tensor is guaranteed to satisfy G^mn_;n = 0.

 

[Bill_K]

No, gravitational waves are very well understood by now, both theoretically and computationally. Including nonlinear effects, and coupling to a time-varying source. The linearized solutions look like electromagnetic waves only in the sense that they obey the wave equation. Apparently the derivations you've read are oversimplified.

 

[Bill_K]

What I've never seen is a solution for, say, an oscillating mass that radiates gravitationally. I'm not sure if such a solution can exist because the EMT of the source would not be conservative, whereas the Einstein tensor is guaranteed to satisfy G^mn_;n = 0.

Exact solutions tend to be unrealistic. The solution to the type of situation you describe will need to be done numerically. Several groups have now computed the collision of two black holes, including the gravitational radiation that is emitted. See http://www.nasa.gov/vision/universe/starsgalaxies/gwave.html.

 

[TrickyDicky]

I found the derivation of GWs from the linearized theory somewhat unconviving, but there are exact (vacuum) solutions of the field equations that look like plane-fronted waves with parallel rays (pp-waves). These solutions have Riemann curvature associated with the waves.

Interesting, do you have any reference on those solutions?

 

[Bill_K]

Interesting, do you have any reference on those solutions?

Wikipedia has an article on them. As it says there, they were first written down by Hans Brinkmann in 1925.

 

[George Jones]

Interesting, do you have any reference on those solutions?

Chapters 17 - 21 of Exact Space-Times in Einstein's General Relativity by Griffiths and Podolsky,

https://www.amazon.com/dp/0521889278/?tag=pfamazon01-20.

 

[TrickyDicky]

No, gravitational waves are very well understood by now, both theoretically and computationally. Including nonlinear effects, and coupling to a time-varying source.

This doesn't even try to answer in the context of what I posted.
What would you say about the background static spacetime being incompatible with gravitational radiation?

The linearized solutions look like electromagnetic waves only in the sense that they obey the wave equation.

Well. I'd say the fact that they should propagate precisely at c, is another point to consider when referring to the similarity. But certainly they need not be EM waves. Still, since GW are so well understood by now, would you explain to me what exactly is oscillating in gravitational waves? curvature?

 

[TrickyDicky]

Chapters 17 - 21 of Exact Space-Times in Einstein's General Relativity by Griffiths and Podolsky,

https://www.amazon.com/dp/0521889278/?tag=pfamazon01-20.

Thanks, still as Bill K has just pointed out exact solutions tend to be unrealistic, and I would add that those solutions referred to in the Podolsky book are not very realistic.
What this tells us is that being an exact solution of the EFE doesn't guarantee anything, as there are many unphysical solutions that have nothing to do with our universe.

 

[George Jones]

For instance, if the linearized EFE look like EM wave equations why not consider them EM radiation to begin with?

Helicity. The polarization states of gravitational and electromagnetic radiation transform differently under rotations.

 

[Nabeshin]

Several groups have now computed the collision of two black holes, including the gravitational radiation that is emitted. See http://www.nasa.gov/vision/universe/starsgalaxies/gwave.html.

I'd like to emphasize this. There's really no doubt that Einstein's GR does contain gravitational waves. When we actually code in his equations and evolve the binary black hole spacetime, we get very clearly GW coming out. This is something I look at on a daily basis, so I'm quite confident in it

 

[Bill_K]

What would you say about the background static spacetime being incompatible with gravitational radiation?

You mentioned Birkhoff's Theorem, but that applies only to spherically symmetric perturbations. Gravitational waves are quadrupole and higher.

Still, since GW are so well understood by now, would you explain to me what exactly is oscillating in gravitational waves? curvature?

Gravitational radiation is characterized by a 1/r term in the Riemann curvature tensor. Large amplitude waves need to be treated numerically, but for example you can talk about small amplitude perturbations of the Schwarzschild solution. In second order one sees that the Schwarzschild mass decreases with time, corresponding to the energy being carried away by the waves.

 

[Mentz114]

I've been reading back copies of 'Matters of Gravity' and there's a fairly detailed description of the numerical methods in this spring 2006 issue.

http://www.phys.lsu.edu/mog/pdf/mog27.pdf

Extract:

Simulating binary black holes has been a long-standing problem because it poses a number
of “grand challenges”. An incomplete list of these challenges includes the following
• Einstein’s equations form a complicated, coupled set of non-linear PDEs, and it is far
from clear which form of these equations is most suitable for numerical implementation.
• Somewhat related is the coordinate freedom, and the question what coordinate (or
gauge) conditions lead to a non-pathological evolution.
• Black holes contain singularities, which can have very unfortunate consequences for
numerical simulations.
• The individual black holes are much smaller than the wavelength of the emitted gravitational
radiation. The resulting range in length-scales is difficult to accommodate in
numerical simulations.

 

[TrickyDicky]

You mentioned Birkhoff's Theorem, but that applies only to spherically symmetric perturbations. Gravitational waves are quadrupole and higher.

That's what I'm saying, that there seems to be some incompatibility between a background spacetime that is spherically symmetric and the fact that this kind of spaces don't admit GW kind of perturbations. But rephrasing it doesn't help me understand it.

Gravitational radiation is characterized by a 1/r term in the Riemann curvature tensor. Large amplitude waves need to be treated numerically, but for example you can talk about small amplitude perturbations of the Schwarzschild solution. In second order one sees that the Schwarzschild mass decreases with time, corresponding to the energy being carried away by the waves.

Again, how can the Schwarzschild spacetime, which is also spherically symmetric and static say anything about GW? It's so confusing.

 

[TrickyDicky]

Helicity. The polarization states of gravitational and electromagnetic radiation transform differently under rotations.

I know GW are postulated to have different helicity than EM waves, in fact the boson proposed for the grav. radiation, the graviton (that BTW hasn't been detected) has spin 2 in contrast with the spin 1 of the EM radiation boson (the photon) so it is straightforward that they should have different helicity.
The issue here would be how does that postulated difference translate to observational tests, and there one faces the problem that gravitational waves (or gravitons for that matter) have not been directly detected so far, and according to some people they might not be detected in many years if ever, so this raises problems with falsability, since the indirect observations that makes us suspect the existence of GW (orbital decay of certain binary sistems) can't distinguish helicity differences.

 

[Mentz114]

... but for example you can talk about small amplitude perturbations of the Schwarzschild solution. In second order one sees that the Schwarzschild mass decreases with time, corresponding to the energy being carried away by the waves.

Do the perturbations, in this case, destroy the spherical symmetry ? Like a short-lived dimple ?

 

[TrickyDicky]

I've been reading back copies of 'Matters of Gravity' and there's a fairly detailed description of the numerical methods in this spring 2006 issue.

http://www.phys.lsu.edu/mog/pdf/mog27.pdf

Extract:

Thanks, at least someone else acknowledges that all is not so nice and easy with the BH binaries numerical simulations as Bill k and Nabeshin seem to imply

 

[PAllen]

That's what I'm saying, that there seems to be some incompatibility between a background spacetime that is spherically symmetric and the fact that this kind of spaces don't admit GW kind of perturbations. But rephrasing it doesn't help me understand it.


Again, how can the Schwarzschild spacetime, which is also spherically symmetric and static say anything about GW? It's so confusing.

Any gravitational wave solution is not spherically symmetric or static, so you can't talk about gravitatational waves in the Schwarzschild spacetime; they are ruled out by definition. This solution is for a single static mass in an empty universe - not very interesting. It's like saying 'how can you have EM waves in a Maxwell solution for a single, stationary charge?"

 

[TrickyDicky]

Any gravitational wave solution is not spherically symmetric or static, so you can't talk about gravitatational waves in the Schwarzschild spacetime; they are ruled out by definition. This solution is for a single static mass in an empty universe - not very interesting. It's like saying 'how can you have EM waves in a Maxwell solution for a single, stationary charge?"

Good point, and nobody seems to be aware of this.

 

[Nabeshin]

Thanks, at least someone else acknowledges that all is not so nice and easy with the BH binaries numerical simulations as Bill k and Nabeshin seem to imply

This is from 2006, around the time when Pretorius made his breakthrough involving constraint damping parameters for numerical relativity. The field has advanced considerably since then, to the point where I feel like most groups doing numerical relativity have conquered many of the large issues quoted.

Extraction and analysis of gravitational wave signals from a binary inspiral is now quite routine (to the point where an undergraduate, namely myself, can do it).

I don't mean to imply that there are no difficulties with numerical relativity, just that extracting gravitational wave signals is not a very difficult problem. One problem encountered is that the waves are necessarily evolved on a finite grid and thus subject to 1/r gauge effects. But this isn't a big deal since there are schemes for extrapolating this out to infinity or using CCE to get the 'true' waveform. Similarly, there are all kinds of difficulties in simulating dynamical spacetimes and getting meaningful information once one has the waves, but I don't think the waves themselves are too difficult.

 

[WannabeNewton]

I don't see how you could interpret the wave equation for GW as possibly being EM waves when the linear EFEs in vacuum involve differential equations with a symmetric, second - rank tensor as opposed to EM's 4 - potential.

 

[cosmik debris]

Still, since GW are so well understood by now, would you explain to me what exactly is oscillating in gravitational waves? curvature?

The Riemann tensor describes the curvature and has two components, the Ricci tensor and the Weyl tensor. The Weyl tensor carries the changes in curvature.

 

[TrickyDicky]

I don't see how you could interpret the wave equation for GW as possibly being EM waves when the linear EFEs in vacuum involve differential equations with a symmetric, second - rank tensor as opposed to EM's 4 - potential.

You do know there is a tensorial formulation of electromagnetism, don't you? The electromagnetic tensor is antisymmetric but remember that in the derivation of GW from the linearized EFE a Lorenz (aka "de donder") gauge condition is imposed that symmetrizes the tensor.On the other hand, I'm not asserting that the waves derived from the linearized EFE have to be EM waves, but merely suggesting that interpreting them as a new form of radiation is just a model-dependent interpretation of GR, that is not without a number of theoretical problems that have not been fully solved and yes, so far it appears as the most correct, but this might be just because there is no other model around to fit a different interpretation in.

The main problem I pointed out was a problem of coherence that has not been satisfactorily answered so far, that of deriving gravitational waves from spacetime models that don't admit that kind of radiation.
After reading some more, there seems to be different approaches to this problem, for instance the "Bondi radiation coordinates" and similar that deal with things like a pulsating kind of universe that alternates from a static configuration when there is no GW perturbation and a different configuration when the perturbation is produced. Not very physical kind of scenarios apparently, but maybe I'm not understanding them well.
Perhaps somebody is more versed in these and could explain them to me.

 

[TrickyDicky]

The Riemann tensor describes the curvature and has two components, the Ricci tensor and the Weyl tensor. The Weyl tensor carries the changes in curvature.

This seems to be unrelated to my question, I asked what oscillates in a gravitational wave?

 

[TrickyDicky]

For instance when it is said that GW are ripples of curved spacetime, I find very difficult to picture it, because one would think that it would have to ripple wrt something, if the ripple describes some kind of motion-like process, how can curved spacetime move?
In the case of EM waves there is a fixed background space with respect to which one can say waves propagate, but when what propagates is the very curved spacetime is hard to see what can be the reference for that motion.

Maybe someone can clarify this point?

 

[PAllen]

For instance when it is said that GW are ripples of curved spacetime, I find very difficult to picture it, because one would think that it would have to ripple wrt something, if the ripple describes some kind of motion-like process, how can curved spacetime move?
In the case of EM waves there is a fixed background space with respect to which one can say waves propagate, but when what propagates is the very curved spacetime is hard to see what can be the reference for that motion.

Maybe someone can clarify this point?

Picture a balloon filled loosely. Tap it and wave motion will propagate. To a 2-d being living in the balloon surface, you have ripples of geometric distortion. That the balloon is embedded in 3-space in inconsequential (esp. for 2-d being). Generalize this analogy to 4-d.

Also, I seem to recall MTW has a long section on how an array of spaceships would perceive a passing gravitational wave. I don't have time right now to find the section number.

 

[TrickyDicky]

Picture a balloon filled loosely. Tap it and wave motion will propagate. To a 2-d being living in the balloon surface, you have ripples of geometric distortion. That the balloon is embedded in 3-space in inconsequential (esp. for 2-d being). Generalize this analogy to 4-d.

There are several problems I have to generalize it to 4-d:

First, the perturbation in this picture comes from the ambient 3-space and needs an ambient time but in the real case an ambient 5-space is not the source of the perturbation, instead the perturbation arises within the 4-spacetime manifold.

Also the main difference with an actual metric (curved 4-spacetime) perturbation is that in the case of the balloon the wave motion propagates as a function of ambient time that is not part of the perturbation but a parameter, so the 2-d beings living in the surface are able to perceive the tap because time is not included in the geometrical perturbation but is the reference wrt which the motion is perceived.
The same happens when we perceive a sismic wave, which is 3-d wave motion wrt time, not a 4-d perturbation like in the case of GW where is curved spacetime what is supposed to propagate.

 

[cosmik debris]

This seems to be unrelated to my question, I asked what oscillates in a gravitational wave?

The curvature as expressed by the Weyl tensor.

 

[TrickyDicky]

The curvature as expressed by the Weyl tensor.

Right, that is where my doubt enters, since curvature includes the spacetime, with respect to what does curvature oscillate? It would seem as if another dimension was needed as reference for spacetime curvature to oscillate. Or how else would we notice that the geometry of our universe (the curvature) is oscillating?
For instance, we notice that the universe is expanding because it is only the spatial part that is expanding wrt time. If spacetime (both space and time) expanded we wouldn't be able to notice.

 

[Q-reeus]

Right, that is where my doubt enters, since curvature includes the spacetime, with respect to what does curvature oscillate? It would seem as if another dimension was needed as reference for spacetime curvature to oscillate. Or how else would we notice that the geometry of our universe (the curvature) is oscillating?..

Just a layman on this topic, but isn't it so curvature is detectable (in principle - there is no direct proof to date) as effect of gradient, not as 'absolute' value? My main problem with GW's in GR setting is the absurdity imo of there being no assigned value for gravitational energy density in the case of a static gravitating mass, but as a freely propagating disturbance, gravity 'magically' acquires energy density, a la binary pulsar data and it's interpretation. Where is consistency here?

For instance, we notice that the universe is expanding because it is only the spatial part that is expanding wrt time. If spacetime (both space and time) expanded we wouldn't be able to notice.

Unless I have it completely wrong, that is not true. Hubble redshift is equivalent to saying the universe was 'slower' (lower clock rate) back then relative to now, no?

 

[TrickyDicky]

Just a layman on this topic, but isn't it so curvature is detectable (in principle - there is no direct proof to date) as effect of gradient, not as 'absolute' value?

Curvature is detectable as gravitational effects (tidal forces) , that is by differentials of relative accelerations.
Ultimately the curvature is due to the spacetime metric evolution, more specifically to the second derivative of metric components that can be viewed as potentials that have gradients whose effects can be detected as curvature (gravity).

My main problem with GW's in GR setting is the absurdity imo of there being no assigned value for gravitational energy density in the case of a static gravitating mass, but as a freely propagating disturbance, gravity 'magically' acquires energy density, a la binary pulsar data and it's interpretation. Where is consistency here?

That is usually dealt with by saying that mostly quadrupole energy density of the orbit system propagates as GW because spherically symmetric perturbations don't generate GW.

Unless I have it completely wrong, that is not true. Hubble redshift is equivalent to saying the universe was 'slower' (lower clock rate) back then relative to now, no?

It is not exactly like saying that back then was "slower" because to do that we would have to compare time rates "now" which is not a well posed comparison in relativity, but actually the Hubble redshift can be viewed as comparison of clock ticking rates in the gravitational redshift interpretation with the adequate coordinates.
In the mainstream most common interpretation of redshift as expansion, what expands is the spatial part of the manifold because if both space and time expanded in the same proportion it would amount to not expanding at all.

 

[Q-reeus]

That is usually dealt with by saying that mostly quadrupole energy density of the orbit system propagates as GW because spherically symmetric perturbations don't generate GW.

Sure, but conditions for mass to be a source of GW's was not the question. Rather, as assumed by there being zero contribution to the energy-momentum stress tensor, spacetime curvature owing to a static mass has no energy content, what is the justification for assigning a 'well defined' value when that curvature is owing to a GW? There is curvature, but then again there is curvature?!

In the mainstream most common interpretation of redshift as expansion, what expands is the spatial part of the manifold because if both space and time expanded in the same proportion it would amount to not expanding at all.

Is there not a distinction between 'space expanding' as meaning increasing total volume of universe (horizon expands and large scale structures move further apart), as opposed to saying only the spatial part of spacetime curvature evolves with time? Not sure whether assumed flatness of universe invalidates the analogy, but seems to me there is one between that situation and the fact that both clock rate and length (radial component only in Schwarzschild metric) are reduced nearer to a gravitating mass.

 

[bcrowell]

Right, that is where my doubt enters, since curvature includes the spacetime, with respect to what does curvature oscillate? It would seem as if another dimension was needed as reference for spacetime curvature to oscillate. Or how else would we notice that the geometry of our universe (the curvature) is oscillating?

Gravitational waves are no different in this respect than any other form of spacetime curvature. You don't need to embed the four-dimensional spacetime in a five-dimensional spacetime in order to have curvature.

 

[bcrowell]

Just a layman on this topic, but isn't it so curvature is detectable (in principle - there is no direct proof to date) as effect of gradient, not as 'absolute' value? My main problem with GW's in GR setting is the absurdity imo of there being no assigned value for gravitational energy density in the case of a static gravitating mass, but as a freely propagating disturbance, gravity 'magically' acquires energy density, a la binary pulsar data and it's interpretation. Where is consistency here?

It's not magic, it's mathematics. Arnowitt, Deser and Misner proved that the ADM energy is conserved in asymptotically flat spacetimes. Also, it's straightforward to prove that GR can't have a general law of conservation of energy that applies to all spacetimes (see MTW, p. 457). If you think there is a lack of consistency, then apparently you believe that one of these proofs has a mistake in it. In that case, you should publish your refutation.

 

[WannabeNewton]

It's not magic, it's mathematics. Arnowitt, Deser and Misner proved that the ADM energy is conserved in asymptotically flat spacetimes. Also, it's straightforward to prove that GR can't have a general law of conservation of energy that applies to all spacetimes (see MTW, p. 457). If you think there is a lack of consistency, then apparently you believe that one of these proofs has a mistake in it. In that case, you should publish your refutation.

Does the lack of a general conservation law for energy have to do with space - time being a manifold that is not embedded in a higher manifold so there is no way to define global energy for space - time (in GR)?

 

[Q-reeus]

It's not magic, it's mathematics. Arnowitt, Deser and Misner proved that the ADM energy is conserved in asymptotically flat spacetimes. Also, it's straightforward to prove that GR can't have a general law of conservation of energy that applies to all spacetimes (see MTW, p. 457). If you think there is a lack of consistency, then apparently you believe that one of these proofs has a mistake in it. In that case, you should publish your refutation.

Made it clear I'm a layman in GR, so obviously there is no intention of publishing some 'refutation' - just want justification and clarification. Perhaps you could explain in a simplified but adequate manner the problem as I see it in #32? Just to make it clear - it's not about whether conservation of energy holds in GR in general, I can live with that. it's that curvature is attributed zero energy density in the static case, yet a non-zero value in the dynamic case. Not a matter of degree but of kind. How and why?

 

[bcrowell]

Does the lack of a general conservation law for energy have to do with space - time being a manifold that is not embedded in a higher manifold so there is no way to define global energy for space - time (in GR)?

No.

If you want to understand why it is, maybe you could say a little about your math and physics background. The MTW reference I gave in #34 explains it, but that will only work for you if you have enough background to understand MTW (Misner, Thorne, and Wheeler, Gravitation).

 

[bcrowell]

Made it clear I'm a layman in GR, so obviously there is no intention of publishing some 'refutation' - just want justification and clarification. Perhaps you could explain in a simplified but adequate manner the problem as I see it in #32?

Could you say something about your background in math and science? Then I could recommend what to start reading.

[...]curvature is attributed zero energy density in the static case, yet a non-zero value in the dynamic case.

This isn't correct.

 

[WannabeNewton]

No.

If you want to understand why it is, maybe you could say a little about your math and physics background. The MTW reference I gave in #34 explains it, but that will only work for you if you have enough background to understand MTW (Misner, Thorne, and Wheeler, Gravitation).

I have the book. I just don't understand the wording involved. It seems to be saying that there is no reference with which to measure global quantities like the total angular momentum...

 

[Q-reeus]

Could you say something about your background in math and science? Then I could recommend what to start reading.

No science background (as in career), but enough maths, largely self-taught, to understand vector algebra and rudiments of vector calculus. But I was hoping for an explanation here, not a referral to some tome there.

This isn't correct.

You may be right, but here's one place, referred to quite often it seems, I took my que from:
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html
"One other complaint about the pseudo-tensors deserves mention. Einstein argued that all energy has mass, and all mass acts gravitationally. Does "gravitational energy" itself act as a source of gravity? Now, the Einstein field equations are

Gmu,nu = 8pi Tmu,nu

Here Gmu,nu is the Einstein curvature tensor, which encodes information about the curvature of spacetime, and Tmu,nu is the so-called stress-energy tensor, which we will meet again below. Tmu,nu represents the energy due to matter and electromagnetic fields, but includes NO contribution from "gravitational energy". So one can argue that "gravitational energy" does NOT act as a source of gravity. On the other hand, the Einstein field equations are non-linear; this implies that gravitational waves interact with each other (unlike light waves in Maxwell's (linear) theory). So one can argue that "gravitational energy" IS a source of gravity."

Which is why I thought someone here might know, because there is no solution offered above - just a brief acknowledgment of the problem.

 

[bcrowell]

I have the book. I just don't understand the wording involved. It seems to be saying that there is no reference with which to measure global quantities like the total angular momentum...

Yep, that's basically it. In relativity, energy is the timelike component of a four-vector, so you basically can't represent it in a global way, because there is no globally defined frame of reference in which to express it.

 

[bcrowell]

No science background (as in career), but enough maths, largely self-taught, to understand vector algebra and rudiments of vector calculus. But I was hoping for an explanation here, not a referral to some tome there.

You can't understand a difficult subject like GR without reading books.

 

[bcrowell]

For anyone here who might be interested, I've written a FAQ entry about what books to read in order to learn GR.

FAQ: I want to learn about general relativity. What books should I start with?

The following is a list of books that I would recommend, sorted by the level of presentation. I've omitted many excellent popular-level books that aren't broad introductions to GR, as well as classic books like Wald and MTW that are now many decades out of date. Before diving into any of the GR books that are aimed at physics students, I would suggest preparing yourself by reading a good textbook on SR such as Taylor and Wheeler, "Spacetime Physics."

Books using only algebra, trig, and geometry:

Gardner, "Relativity Simply Explained"

Einstein, "Relativity: The Special and General Theory ," http://etext.virginia.edu/toc/modeng/public/EinRela.html

Geroch, "General Relativity from A to B"

Will, "Was Einstein Right?"

Books assuming a lower-division university background in math and physics:

Taylor and Wheeler, "Exploring Black Holes: Introduction to General Relativity"

Hartle, "Gravity: An Introduction to Einstein's General Relativity"

Rindler, "Relativity: Special, General, and Cosmological"

Books for grad students in physics:

Carroll, "Spacetime and Geometry: An Introduction to General Relativity," available for free online in an earlier and less complete form at http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll_contents.html

 

[TrickyDicky]

Gravitational waves are no different in this respect than any other form of spacetime curvature. You don't need to embed the four-dimensional spacetime in a five-dimensional spacetime in order to have curvature.

Sure, you don't need it to have curvature but having curvature seems to be different from having oscillations of curvature in the form of waves. Having curvature explains why 3-d objects fall down or why they orbit each other as they follow geodesics, but here we are talking about a type of waves in which what oscillates is the 4-d spacetime curvature, surely to assert that something that is 4d is oscillating we need to invoke a fifth dimension, just like to conceive ondulatory motion of 3d objects we need a 4th parameter (time dimension) or we don't have waves at all. This works for any number n of dimensions, i.e. if we want a just x or y dimension harmonic oscillator we need a second dimension (time) to have periodic motion.

 

[bcrowell]

Sure, you don't need it to have curvature but having curvature seems to be different from having oscillations of curvature in the form of waves. Having curvature explains why 3-d objects fall down or why they orbit each other as they follow geodesics, but here we are talking about a type of waves in which what oscillates is the 4-d spacetime curvature, surely to assert that something that is 4d is oscillating we need to invoke a fifth dimension, just like to conceive ondulatory motion of 3d objects we need a 4th parameter (time dimension) or we don't have waves at all. This works for any number n of dimensions, i.e. if we want a just x or y dimension harmonic oscillator we need a second dimension (time) to have periodic motion.

I'm sure this seems self-evident to you, but it's not true. I would suggest that you do some reading from the list of books I posted in #43. If you get yourself up to the level of the Rindler book, he gives a very nice presentation of gravitational waves.

 

[TrickyDicky]

I'm sure this seems self-evident to you, but it's not true. I would suggest that you do some reading from the list of books I posted in #43. If you get yourself up to the level of the Rindler book, he gives a very nice presentation of gravitational waves.

Honestly, that response looks like you are avoiding answering the very clear and simple set up of my post.

 

[Q-reeus]

You can't understand a difficult subject like GR without reading books.

Thanks for the textbook links in #43, but granting the above, I'm not out to master GR. Many people at this forum ask all sorts of 'dumb-to-smart' questions and tend to get helpful and specific answers. So are you saying there is no reasonably simple way of explaining the problem I have outlined in #30,32,36,40? If you say the question itself is wrong, how do you understand the passage I reproduced in #40? Seems to me it's a case of joining the dots, which in this setting results in two skew lines - on the surface at least there is a consistency problem. Can't see a basic explanation needing several truckloads of high level maths, but hell I could be wrong!

 

[TrickyDicky]

Just a layman on this topic, but isn't it so curvature is detectable (in principle - there is no direct proof to date) as effect of gradient, not as 'absolute' value? My main problem with GW's in GR setting is the absurdity imo of there being no assigned value for gravitational energy density in the case of a static gravitating mass, but as a freely propagating disturbance, gravity 'magically' acquires energy density, a la binary pulsar data and it's interpretation. Where is consistency here?

Sure, but conditions for mass to be a source of GW's was not the question. Rather, as assumed by there being zero contribution to the energy-momentum stress tensor, spacetime curvature owing to a static mass has no energy content, what is the justification for assigning a 'well defined' value when that curvature is owing to a GW? There is curvature, but then again there is curvature?!

it's that curvature is attributed zero energy density in the static case, yet a non-zero value in the dynamic case. Not a matter of degree but of kind. How and why?

I understand now what you meant in posts #30 and #32, and I completely agree with you that there is a major inconsistency there unless someone explains some hidden assumptions we might be missing.
In a way this is somewhat related to the distinction I make between curvature and oscillations of curvature but in your case you are making the distinction between no defined gravity energy due to the curvature of a particular mass and well defined value for the gravitational energy of a gravitational wave (oscillation of curvature).

You may be right, but here's one place, referred to quite often it seems, I took my que from:
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html
"One other complaint about the pseudo-tensors deserves mention. Einstein argued that all energy has mass, and all mass acts gravitationally. Does "gravitational energy" itself act as a source of gravity? Now, the Einstein field equations are

Gmu,nu = 8pi Tmu,nu

Here Gmu,nu is the Einstein curvature tensor, which encodes information about the curvature of spacetime, and Tmu,nu is the so-called stress-energy tensor, which we will meet again below. Tmu,nu represents the energy due to matter and electromagnetic fields, but includes NO contribution from "gravitational energy". So one can argue that "gravitational energy" does NOT act as a source of gravity. On the other hand, the Einstein field equations are non-linear; this implies that gravitational waves interact with each other (unlike light waves in Maxwell's (linear) theory). So one can argue that "gravitational energy" IS a source of gravity."

Which is why I thought someone here might know, because there is no solution offered above - just a brief acknowledgment of the problem.

Yes, here the problem is clearly acknowledged but then ignored. I would tend to think that if this problem is not solved in GR, and as it is shown in this thread, it is not, rather it's ignored, it is at least debatable whether GW have a solid theoretical base.

 

[Q-reeus]

I understand now what you meant in posts #30 and #32, and I completely agree with you that there is a major inconsistency there unless someone explains some hidden assumptions we might be missing.
In a way this is somewhat related to the distinction I make between curvature and oscillations of curvature but in your case you are making the distinction between no defined gravity energy due to the curvature of a particular mass and well defined value for the gravitational energy of a gravitational wave (oscillation of curvature)...

Exactly TrickyDicky. And it gets even worse if one takes seriously the very popular belief amongst cosmologists that the total energy of the universe is zero, which owing to the massive amount of positive energy tied up mainly in matter, requires gravity to take on an equally massive negative value in this setting. Sure seems like a chameleon; positive, negative, zero - take your pick! Not trying to be flippant about this, but such an apparent ability to change sign surely goes way beyond simply being ill-defined in the large.

Yes, here the problem is clearly acknowledged but then ignored. I would tend to think that if this problem is not solved in GR, and as it is shown in this thread, it is not, rather it's ignored, it is at least debatable whether GW have a solid theoretical base.

It's disappointing that with any number of GR buffs on this forum no-one has so far proffered some definite answer, but I suppose folks have their reasons.
There are for me other serious GR consistency issues I shouldn't go into here, but in searching around have found two distinctly different alternatives to GR that at least look like providing some resolution, but from different perspectives. I won't post links because that will be inviting immediate censure, but if you want to you can try web searching using "Huseyin Yilmaz" (has a metric theory very similar to GR but where a definite energy density is ascribed to curvature), or Yuri Baryshev (proponent of a field theory where again gravity has a well defined energy density). The latter has I suppose a real problem cosmology wise in that it doesn't seem to admit to a Big Bang, but I'm not 100% on that.

Just on the matter of 4d ripples. Only my rather simple way of looking at this analogy wise, but is not a sonar beam propagating through say water a (3+1)d disturbance within a (3+1)d continuum? Perhaps quite inapt, but I tend to think of the pressure as a substitute for 4-space curvature, and pressure gradient = flow rate as substitute for 'tidal forces', with the massive caveat that sonar beam is a longitudinal monopolar wave, while GW is transverse quadrupolar.
EDIT: Better analogy might be transverse shear waves propagating through a solid - but of course giving only spatial distortions as a function of time.

 

[TrickyDicky]

It's disappointing that with any number of GR buffs on this forum no-one has so far proffered some definite answer, but I suppose folks have their reasons.

Can't think of any reason someone with knowledge might refuse to try and answer this questions in a forum that is devoted to do exactly that.

There are for me other serious GR consistency issues I shouldn't go into here, but in searching around have found two distinctly different alternatives to GR that at least look like providing some resolution, but from different perspectives.

I 'm convinced this issues have an appropriate solution within GR. It's a matter of time.


To comeback to the topic proper, I think there is a close relation between the almost a century long debates about conservation of energy-momentum in GR and the issues about gravitational radiation, for instance in the specific case of the binary pulsar (Hulse-Taylor pulsar), when we interpret the shrinking of the binary system orbit as energy lost by emission of gravitational radiation, we are relying on the fact that in GR there is no global energy-momentum conservation (at least for the quadrupole momentum)-if this is not correct please somebody correct me.
But in an imaginary scenario with global energy conservation the orbital decay of exactly the amount dictated by the quadrupole moment tensor(plus higher order negligible multipoles) for the masses and eccentrity of the orbit, would come imposed just by angular momentum conservation considerations (Noether theorem).
That is one motive why in a static spacetime GW are not to be found, but then again a static spacetime is not a cosmology that probably admits binary systems.