Doubt about gravitational waves

[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.

 

[Q-reeus]

...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)...

Can't quite follow that bit - isn't it the case we are relying on conservation of energy-momentum to explain the fit between data and theory?
The striking fit of that Hulse-Taylor pulsar data to theory convinces me that for sure GW's exist and carry positive energy. For other situations things make much less sense, unless somehow there is a complete break from the (approximate) quadrature dependence between 'stress' and energy density for both static and radiative fields that sensibly applies to EM, acoustics etc. I can't see how there could be consistency, but maybe someone here knows. Or not. Just came across the following perfectly kosher presentation of GW physics that may answer at least some of your questions, but honestly your best bet may be contacting someone like Kip Thorne direct:
The Physics of Gravitational Waves and their Generation - K.Thorne http://www.ilorentz.org/lorentzchair/thorne/Thorne1.pdf (really starts p12, p14 contains a minor gaff re 'spin' formula)
Going through it, but still can't see where background (ie static) curvature is assigned some energy density.

 

[cosmik debris]

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.

Their is no need to embed anything to notice a change in curvature. Gravitational radiation will distort the shape of an object as it passes through. In simple terms the Ricci tensor dictates how the volume of a bunch of test particles changes and the Weyl tensor how the shape changes.

 

[TrickyDicky]

Can't quite follow that bit - isn't it the case we are relying on conservation of energy-momentum to explain the fit between data and theory?

There is an interesting thread on this global conservation issue: https://www.physicsforums.com/showthread.php?t=490368

 

[cosmik debris]

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.

Ummm, basically no. EM waves are are described in 4D spacetime using the Faraday Tensor without any need for another dimension. In 1, 2, and 3D oscillations time is a parameter not another dimension. The beauty of relativity was the incorporation of time as a dimension.

 

[TrickyDicky]

EM waves are are described in 4D spacetime using the Faraday Tensor without any need for another dimension.

More accurately what the EM tensor describes is the EM field in time, and the time varying evolution of this type of fields are naturally described within a 4D spacetime. This is not in contradiction with what I wrote. But I'm sure you'll agree that the field that oscillates in a EM wave is 3D(spatial) field and therefore we need a 4D (n+1) description (as you point out) of the phenomenon, just what I'm trying to get across.
However in a GW what oscillates is 4D to begin with (because time is included in what is propagating:the spacetime geometry of our universe), so it seems natural that to detect it we'd need one more dimension.

 

[Q-reeus]

Their is no need to embed anything to notice a change in curvature. Gravitational radiation will distort the shape of an object as it passes through. In simple terms the Ricci tensor dictates how the volume of a bunch of test particles changes and the Weyl tensor how the shape changes.

That nice simple summary appeals to me. Maybe wrong here, but unless one assumes a priori an infinite and perfectly flat or at least uniform nature, does not requiring a higher imbedding dimension invite an infinite succession - the higher imbedding dimension in general having some structure (curvature) which in turn requires a yet higher imbedding dimension to define, and so on?
Too bad no-one has a similar simple suggestion to resolve the 'chameleon' energy problem suggested earlier in this thread.

 

[Q-reeus]

There is an interesting thread on this global conservation issue: https://www.physicsforums.com/showthread.php?t=490368

Thanks for the lead. The issue there is similar, but differs in that illdefinedness of global energy in GR is the key issue - the problems raised in this thread are somewhat more stark. Notable though that one respondent there offers, straight off the bat, reasonably detailed explanations, yet on this thread queries one's credentials first before a curt referral to some textbook(s). An 'interesting' contrast in style, given the close similarity in content of both threads.

 

[TrickyDicky]

Thanks for the lead. The issue there is similar, but differs in that illdefinedness of global energy in GR is the key issue - the problems raised in this thread are somewhat more stark.

Yeah, people tend to speak more freely in abstract or general terms, but when going to specific or controversial examples the fear to say something that might contradict the official doctrine is very strong around here.

Notable though that one respondent there offers, straight off the bat, reasonably detailed explanations, yet on this thread queries one's credentials first before a curt referral to some textbook(s). An 'interesting' contrast in style, given the close similarity in content of both threads.

Curious indeed, see above.

 

[Q-reeus]

Yeah, people tend to speak more freely in abstract or general terms, but when going to specific or controversial examples the fear to say something that might contradict the official doctrine is very strong around here.

Can't help but agree (although in my case there's also a 'prior history' factor)! There's a kind of no-mans land here at PF imo. In this and similar sections, all sorts of weird/dumb opinions can be raised initially by people with no maths/physics background at all, and unless particularly belligerent or crazy, such OP is typically treated with respect and attention unless he/she fails to 'sees the light' eventually. Nothing against that in principle -it mostly works fine. At the other end, there is the section 'Beyond the Standard Model' for high end mathematical debates by dedicated theorists with new and detailed theories. Again, fine. It's when one doesn't agree with or strongly queries some aspect of established theory (ie GR or QM), but hasn't the interest or capacity to invent some whole new paradigm that there's a real dilemma. Very easy to tread on toes, and some key players here tend to have long memories! BTW, was that reference to Kip Thorne's article of any use?

 

[Mentz114]

This is an interesting thread and made me think of a point which I'll take the liberty of raising here.

Assuming the curvature of a GW is Weyl, that is shape-changing but not volume changing this puts some constraints on the tidal (gravitoelectric) tensor, which is given by the spatial part of this tensor (evaluated in the local frame, so the indexes a,b,c,d are frame indexes not holonomic)

<br /> T_{ac} = R_{abcd}\ u^b\ u^d<br />

in the frame of an observer with four velocity u. So in the local frame we take u^0=-1,\ \ u^k = 0,\ \ k=1,2,3. Suppose we are working in rectilinear coords t,x,y,z and a GW passes in the z-direction. Wouldn't the tidal tensor then take the form diag(w_x(\vec{x},t,\lambda),w_y(\vec{x},t, \lambda),0) ?

The symmetry demands that the x- and y- tidal effects must be the same but out of phase (spatially) by \lambda/2, and that w_x+w_y=0 to reflect no change in volume. This tidal tensor will cause the squishing/stretching effect postulated for GWs ( I could be wrong on this point. Maybe some cross-terms are required because of the phase).

If this is possible then we have a wave of purely spatial nature evolving with t as the parameter, just like EM waves.