I Gravitational Waves & Matter: Causes, Effects & Thresholds

[PeterB]

From what I have read gravitational waves are caused by the acceleration of massive object causing ripples in space time. What specifically causes this, and how does general relativity predict these. Does it have to be a high density of matter, or a large amount of it. How do these waves affect the matter and spacetime they travel through, and is there affect lesser with distance traveled? Is there a threshold of where gravitational waves cease to be made?
Good sources on the matter would be appreciated as well, most that I have found tend to be either very rudimentary or in such technical language that I can't get my foot in the door.

 

[Ibix]

From what I have read gravitational waves are caused by the acceleration of massive object causing ripples in space time.

A changing quadropole moment in anything that produces gravity (mass, energy, and stresses such as pressure) produces gravitational waves. This does include accelerating masses, but also orbiting bodies that are in free fall and not accelerating in any physically meaningful sense.

What specifically causes this, and how does general relativity predict these.

Gravitational waves arise for the same reason that electromagnetic waves do. Nothing can travel faster than the speed of light, so if I accelerate a charge (which produces an electromagnetic field) my friend one light second away won't see a change in the electromagnetic field where she is for a second, and my friend two light seconds away won't see a change for two seconds, etc. The propagating change in the electromagnetic field turns out to be an electromagnetic wave. Similarly, if I change a mass distribution my friend a light second away won't see a change in the gravitational field for a second, and the propagating change in the gravitational field is a gravitational wave. Because the gravitational field doesn't have the same properties as the electromagnetic field, the maths is much nastier.

Does it have to be a high density of matter, or a large amount of it.

Any mass will do, but gravity is so weak that the power levels are ridiculously low - the Earth orbiting the Sun emits gravitational waves with a power less than 100W.

How do these waves affect the matter and spacetime they travel through, and is there affect lesser with distance traveled?

There's a good graphic on Wikipedia showing how a ring of inertial masses is affected by a gravitational wave passing through perpendicular to the plane of the ring. The distance in one direction shrinks while the distance in the perpendicular direction grows, then vice versa. Yes, the effect drops off with distance, which is why we need such incredibly sensitive detectors to detect the effect of something as cataclysmic as two large black holes orbiting each other at near the speed of light.

 

[pervect]

Gravitational waves are a prediction of General relativity (henceforth, GR). General relativity is a nonlinear field theory, but it can be linearized, and Gravitational waves can be associated with the linearization of the full theory of GR.

There's a very rough approximate expression for the total radiated power in gravitational radiation in MTW's textbook, "Gravitation." eq 36.5, pg 978. MTW determines that the radiated power is proportional to the square of the internal power flow. In standard units, it's roughly the square of the internal power divided by 4*10^52 watts. That's about 200,000 solar masses (times c^2 to covert to energy) per second.

The equations may not be completely general, but is intended to be applied to typical sources of graitaional waves, like binary inspirals.

As far as good sources go, I would suggest looking at LIGO, the project that detected gravitational waves. The arxiv version of LIGO's first detection paper is at https://arxiv.org/abs/1602.03837. I think Ligo also has the same at https://www.ligo.caltech.edu/system...inal/detection-science-summary.pdf?1455157973

LIGO's home page has a number of educational resources, that I haven't really perused recently. From a quick look, the "educational resources" is probably the best section of their webpage to start with, which is at https://www.ligo.caltech.edu/page/educational-resources

 

[pervect]

Some more explanation of the context of the formula for gravitational radiation I gave earlier would, I think, be useful.

The usual formulation uses the third time derivative of the quadropole moment of a system. But that's quite a mouthful and hard to describe. The formula gives the total gravitational radiation as being proportional to the squared magnitude of this quantity.

MTW offers us an easier to understand guide to interpret what this means. Letting the (reduced) quadrupole moment be denoted by $I_{jk}$, we have

$$
\dddot{I_{jk}} \approx \frac{M \, R^2}{T^3}
$$

where M is the mass of the part of the system that moves
R is the size of the system
T is the time it takes for the mass to move from one side of the system to the other.

Then the power flow is, using the Newtonian approxmianion of m v^2, and omitting the factor of two as this is just a very rough estimation

$$P \approx \frac{M\,\left( \frac{R}{T} \right) ^2 }{T}$$

So the total radiated power is proportional to P^2

The formula in terms of the quardupole moment is given earlier as
$$\left< \frac{1}{5} \,\dddot{I_{jk}} \, \dddot{I_{jk}} \right>$$

So the context of this formulation is we have some part of the system moving from "one side" to the other side, in a repetitive manner, and the formula gives us a rough order-of-magnitude estimation as to the amount of power emitted by gravitational radiation.