Understanding Gravitational Waves & Self-Taught GR

[John_Doe]

Could someone please explain how gravitational waves are modeled within the theory? Is it some sort of time dependent metric, or is it simply an indirect consequence of the theory, etc.?

Also, I am self-taught GR, and I learned almost everything I know about the theory straight out of Einstein's "The Meaning Of Relativity" based on the Princeton lectures. I know that this book is definitely not the best book for learning GR from; are there any concepts that I should revise which Einstein does not cover himself?

 

[robphy]

Some possibly useful reading material [I haven't looked through all of this]:

assorted:
http://arxiv.org/abs/physics/9908041 (introduction)
http://www-conf.slac.stanford.edu/ssi/2005/program.htm (first few days)

Carroll's course notes
http://arxiv.org/abs/gr-qc/9712019

Thorne's course notes:
http://elmer.tapir.caltech.edu/ph237/
http://www.pma.caltech.edu/Courses/ph136/yr2004/ (Ch 26 - backtrack as needed)

 

[member 11137]

Could someone please explain how gravitational waves are modeled within the theory? Is it some sort of time dependent metric, or is it simply an indirect consequence of the theory, etc.?

Also, I am self-taught GR, and I learned almost everything I know about the theory straight out of Einstein's "The Meaning Of Relativity" based on the Princeton lectures. I know that this book is definitely not the best book for learning GR from; are there any concepts that I should revise which Einstein does not cover himself?

Gravitational waves are a direct consequence of the theory (general relativity), more exactly of the fields equations for small variations of the Minkowski metric. This is yielding a linear approximation of these equations that can be considered a little bit like the equations of electrodynamics.

 

[rbj]

Gravitational waves are a direct consequence of the theory (general relativity), more exactly of the fields equations for small variations of the Minkowski metric. This is yielding a linear approximation of these equations that can be considered a little bit like the equations of electrodynamics.

at least a little bit like the equations of electrodynamics. lessee, you have a (pseudo)force that is inverse-square and has action that travels at the speed of c. what set of equations can represent that?

$$\nabla \cdot \mathbf{E} = -4 \pi G \rho \$$

$$\nabla \cdot \left( \frac{1}{2} \mathbf{B} \right) = 0 \$$

$$\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \left( \frac{1}{2} \mathbf{B} \right)} {\partial t} \$$

$$\nabla \times \left( \frac{1}{2} \mathbf{B} \right) = \frac{1}{c} \left( -4 \pi G \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t} \right) = \frac{1}{c} \left( -4 \pi G \rho \mathbf{v}_{\rho} + \frac{\partial \mathbf{E}} {\partial t} \right) \$$

they call these the "gravitoelectromagnetic" (GEM) equations.

For a test particle of small mass, m, the net (Lorentz) force acting on it due to GEM fields is:

$$\mathbf{F}_{m} = m \left( \mathbf{E} + \frac{1}{c} \mathbf{v}_{m} \times \mathbf{B} \right)$$.

looks a helluva lot like Maxwell's Equations to me. (a factor of 1/2 on the B field i never quite understood.) but that should be solvable to a wave equation if need be.

 

[member 11137]

The term "little bit"" was written here in reference to the fact that such similitude is only obtained and valid (so far my knowledges) for the linearized approximation. Until now, I did never read any paper proposing a generalization of this; but I could not read every paper published in this domain, of course. So, it is possible that I missed some important chapters of the modern physics. In this sense, I would say that general solutions of the field equations of the generalised theory of relativity does not automatically lead to a similitude with the behavior of EM waves. Even if I believe that some gravitational waves are parented with some EM waves. Regards

 

[Perturbation]

$$\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2\right) h_{\alpha\beta}=16\pi \frac{G}{c^4}T_{\alpha\beta}$$

Where $h_{\alpha\beta}$ is a linear perturbation of a background Minkowski space-time, descirbing the field of a weak source (or any source at a sufficiently large distance). The above is obtained by the application of various gauge fixing transformations, and using a traceless perturbation, on Einsteins field equations linear in h. (We don't initially need to treat h as a tensor, but a function, however if we do assume there's a background Minkowski space-time the function h automatically transforms as a (0 2) tensor.)

$$g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}$$

$\eta_{\alpha\beta}$ the Lorentzian metric of Minkowski space-time.

 

[rbj]

The term "little bit"" was written here in reference to the fact that such similitude is only obtained and valid (so far my knowledges) for the linearized approximation. Until now, I did never read any paper proposing a generalization of this; but I could not read every paper published in this domain, of course.

someone named Mashoon did some seminal papers in GEM. another named Clark did another paper. with differences in scaling (that was absorbed by the Lorentz force equation), they both came up with something that looked like maxwell's equations given the assumption of reasonably flat space-time.

So, it is possible that I missed some important chapters of the modern physics. In this sense, I would say that general solutions of the field equations of the generalised theory of relativity does not automatically lead to a similitude with the behavior of EM waves.

it can be shown to, in flat space-time.

Even if I believe that some gravitational waves are parented with some EM waves.

it certainly did not surprise me: in the simplistic limit it's an inverse-square action and a speed of action that is c. that doesn't prove anything, but it should hint at it.

 

[member 11137]

someone named Mashoon did some seminal papers in GEM. another named Clark did another paper. with differences in scaling (that was absorbed by the Lorentz force equation), they both came up with something that looked like maxwell's equations given the assumption of reasonably flat space-time..

By side it is a part of the answer to a question that I tried to ask on another forum (sci.physics) concerning the relations between the equations of the field (Einstein) and the MAxwell's laws; and conversely. Thanks

it certainly did not surprise me: in the simplistic limit it's an inverse-square action and a speed of action that is c. that doesn't prove anything, but it should hint at it.

Happy to know that you are not surprised by this assertion.

But can I ask the naive question: if some EM waves are so parented with some gravitational waves, then how can we distinguish them from each other? With other words: is the fact that we are not able (until now) to detect gravitons only correlated with the difficulty of the experiment or for a part connected to the fact that some gravitational waves are transformed into parented EM waves?

Does this question makes any sense accordingly to the actual knowledges ?

Thanks for the time of an answer.

 

[member 11137]

Also, I am self-taught GR, and I learned almost everything I know about the theory straight out of Einstein's "The Meaning Of Relativity" based on the Princeton lectures. I know that this book is definitely not the best book for learning GR from; are there any concepts that I should revise which Einstein does not cover himself?.

I didn't read the book you are using. And I didn't know how far did Einstein go. Some wellknown researchers did continue the work, even when Einstein was still alive. One of them in France was Lichnerowicz. I ignore if you can get a translation in english of this (old book) work but it is a well of informations for me. It is a pleasant feeling to read the developments of a new theory (at this time). One tries to follow the way of thinking like a detective would do it.

Concerning the gravitational waves, your subject, Lichnerowicz and al. could mathematically demonstrate the exact identity (theorem; § 23 ; pages 50-51) between the caracteristic varieties (I hope it is the correct translation of this technical word) involved in both equations: of Einstein and of Maxwell. For him it was the demonstration of a total identity between EM waves and gravitational waves. Furthermore the law of propagation for the EM waves was the same than the law of propagation for gravitational waves. (Théories relativistes de la gravitation et de l'electromagnétisme; Lichnerowicz; Paris ; 1955).

 

[member 11137]

Null length geodesics are gravitational "rays" and are also "EM rays". Any stress energy tensor of any EM field owns 4 eigenvalues (k, k, -k, -k). When k = 0, it is said to be singular (Introducing by side an isotropic vector in the discussion). A singular EM field can be understood as being a fluid of photons following geodesics of null length.

Personal note: This parenty between the two types of waves is naturally telling the question of the behavior of the gravitons in such a fluid... I have no pertinent answer despite all I could read and imagine myself (I know about the difference concerning the helicity and the spin and so on...