Theoretical Understanding of Gravitational Waves

[Isaac0427]

Following the new discovery of gravitational waves by LIGO, I just want to make sure I understand the concept of these waves. I believe I currently have a novice understanding of gravitational waves: when a large, fairly sudden change happens to the position of a particle (acceleration or its derivatives) that has a certain field, waves are created in that field. The waves have properties that reflect on the nature of the field, and therefore according to GR, these waves would be disturbances in spacetime.
I understand that if anything it is an oversimplification, but do I roughly have the idea? Also, could you explain some of the more advanced parts? I know that the Einstein field equations predict gravitational waves, is this just because it shows that gravity causes spacetime curvature? Is there more to it? If yes, could you explain how the EFE predicts them? Thanks guys!

--EDIT--
I apologize for my bad terminology.

 

[andrewkirk]

None of the others, who understand this much better than me, have answered yet, so I'll give it a go.

Broadly your description is correct. I'd just suggest being a bit careful with the phrase 'disturbance in spacetime' because - to me at least - it conjures up an image of spacetime wobbling. But for that to mean anything, we'd need another time dimension in which those wobbles occur, because spacetime already is four-dimensional including a time dimension. A more cautious description would be 'variations over time in the spacetime curvature at a particular point in space'. That description presupposes a coordinate system, in order to make the references to 'time' and a 'point in space' meaningful. But there are standard coordinate systems that can be used to tie that down, such as the 'Swarzschild coordinates'. Another way to think of it is that, if you were a 6-dimensional being standing back and looking at our 4D spacetime embedded in your world, you would see frozen ripple contours in the spacetime, emanating from the wordline of the massive bodies that were generating the grav waves.

As to the more advanced parts, try this link. It explains the generation of gravitational waves. The link is from Schutz's 'A first course in general relativity' - chapter 9 on Gravitational Radiation.

You don't need to fully understand the maths in order to get the general idea. The maths is tensor calculus and differential geometry, which I'm guessing you haven't studied yet. Equation 9.64 is a wave equation derived from the EFE, in a 'linearised form' that uses a pseudotensor $\bar h_{\mu\nu}$ to approximate the Ricci tensor that denotes spacetime curvature in Einstein's equation.

In the region where the wave is being generated there will be big, regular changes in the location of large amounts of mass-energy, such as happens with two large stars or black holes that are orbiting each other in a pair. So at any point within that orbital region, the term on the right of 9.64 $T_{\mu\nu}$, which is the stress-energy tensor and represents the amount of mass-energy at that point, is undergoing regular changes of very large amplitude.

A full solution of the equation gives a formula not only for changes in spacetime curvature within the orbital region, but also arbitrarily far away in space, where - because it is away from the co-orbiting pair of stars - there will be negligible mass-energy so that $T_{\mu\nu}=0$. That solution will involve a sinusoidal variation over time in the spacetime curvature at a spatial point a long way away.

Sorry I can't explain it any better than that right now. I have to brush up on this stuff myself now this discovery has been announced, and working through the maths carefully is quite a slog!

 

[Isaac0427]

None of the others, who understand this much better than me, have answered yet, so I'll give it a go.

Broadly your description is correct. I'd just suggest being a bit careful with the phrase 'disturbance in spacetime' because - to me at least - it conjures up an image of spacetime wobbling. But for that to mean anything, we'd need another time dimension in which those wobbles occur, because spacetime already is four-dimensional including a time dimension. A more cautious description would be 'variations over time in the spacetime curvature at a particular point in space'. That description presupposes a coordinate system, in order to make the references to 'time' and a 'point in space' meaningful. But there are standard coordinate systems that can be used to tie that down, such as the 'Swarzschild coordinates'. Another way to think of it is that, if you were a 6-dimensional being standing back and looking at our 4D spacetime embedded in your world, you would see frozen ripple contours in the spacetime, emanating from the wordline of the massive bodies that were generating the grav waves.

As to the more advanced parts, try this link. It explains the generation of gravitational waves. The link is from Schutz's 'A first course in general relativity' - chapter 9 on Gravitational Radiation.

You don't need to fully understand the maths in order to get the general idea. The maths is tensor calculus and differential geometry, which I'm guessing you haven't studied yet. Equation 9.64 is a wave equation derived from the EFE, in a 'linearised form' that uses a pseudotensor $\bar h_{\mu\nu}$ to approximate the Ricci tensor that denotes spacetime curvature in Einstein's equation.

In the region where the wave is being generated there will be big, regular changes in the location of large amounts of mass-energy, such as happens with two large stars or black holes that are orbiting each other in a pair. So at any point within that orbital region, the term on the right of 9.64 $T_{\mu\nu}$, which is the stress-energy tensor and represents the amount of mass-energy at that point, is undergoing regular changes of very large amplitude.

A full solution of the equation gives a formula not only for changes in spacetime curvature within the orbital region, but also arbitrarily far away in space, where - because it is away from the co-orbiting pair of stars - there will be negligible mass-energy so that $T_{\mu\nu}=0$. That solution will involve a sinusoidal variation over time in the spacetime curvature at a spatial point a long way away.

Sorry I can't explain it any better than that right now. I have to brush up on this stuff myself now this discovery has been announced, and working through the maths carefully is quite a slog!

Thank you very much! I'm working hard on the math, however I will not deny that much of it goes over my head, considering I'm in algebra 2, and all my knowledge of calculus is self-taught.