Undergrad Find GR Equation: Collapsing Orbit & Gravitational Wave

[Buzz Bloom]

TL;DR

Looking for an equation relating two bodies in an orbit which is collapsing is size due to gravitational waves produced by the motion of the planets

I recall some time ago seeing a GR equation describing the rate of orbital energy loss from the moving objects in orbit generating gravitational waves. I can no longer find this equation again. I am hoping someone can help me.

 

[ergospherical]

Define $$I_{ij}(t) = \int_{\Sigma} T^{00}(\tilde{\mathbf{r}}, t) \tilde{x}_i \tilde{x}_j dV$$ where the integration variable $\tilde{\mathbf{r}}$ runs over a spatial volume $\Sigma$ and then write its traceless part as $Q_{ij} = I_{ij} - \frac{1}{3} I_{kk} \delta_{ij}$. The power radiated is$$P(t) = \frac{1}{5} G\frac{d^3 Q_{ij}}{dt^3} \bigg{|}_{t-||\mathbf{r}||} \frac{d^3 Q^{ij}}{dt^3} \bigg{|}_{t-||\mathbf{r}||}$$

 

[Buzz Bloom]

Define $$I_{ij} = \int_{\Sigma} T^{00}(\mathbf{r}, t) x_i x_j dV$$ over a spatial volume $\Sigma$ and then write its traceless part as $Q_{ij} = I_{ij} - \frac{1}{3} I_{kk} \delta_{ij}$. The power radiated is$$P = \frac{1}{5} G \frac{d^3 Q_{ij}}{dt^3} \bigg{|}_{t-||\mathbf{r}||} \frac{d^3 Q^{ij}}{dt^3} \bigg{|}_{t-||\mathbf{r}||}$$

Hi ergospherical:

Your equation looks interesting, but I am not able to interpret its format. Can you possibly please reformat it?

 

[ergospherical]

Buzz Bloom,
Which part of the equation is not clear?

 

[Buzz Bloom]

When I first looked at your post, the format of the equations were unclear on my screen for some unknown reason. I apologize for my confusion. I can now see equations about which I have a few questions.

Is G the constant of Gravity? Are there any assumptions about it's units?

I am guessing V represents a coordinate corresponding to a volume, and the integral is over a volume space Σ. Is this correct? What confuses me it the relationship between Σ and the space involving the orbits of interest.

I get that the integral gives a value for matrix element I_ij based in the factors x_i and x_j. What I do not fully get is the relationship between Σ and the two variables x_i and x_j.

I do not understand the notion of the two vertical bars with t-||r||.

I am unsure of my understanding the subscript k.
I am guessing k = i = j. Is this correct?
If so, then Q_kk = (2/3) I_kk. Is this correct?

I am also guessing that T₀₀ is the element of a tensor corresponding to t,t. Is this correct? Also, can you refer me to a source in which the definition of this T tensor is defined in detail?

Where do the masses of the two objects fit into these equations?

I would much appreciate any help you can give me.

Regards,
Buzz

 

[ergospherical]

Lots of questions!

i) $G$ is the gravitational constant [you can set $G=1$ if you like].
ii) $dV$ is the spatial volume element, e.g. for example simply $dV = dx^1 dx^2 dx^3$ in Cartesians.
iii) $\Sigma$ is some subset of $\mathbb{R}^3$ containing the bodies of interest, over which you do the volume integral.
iv) $||\mathbf{r}|| := \sqrt{(x^1)^2 + (x^2)^2 + (x^3)^2}$ if $\mathbf{r} = (x^1, x^2, x^3)^T$ and $t - ||\mathbf{r}||$ can be called the retarded time.
v) $k$ is summed over, whilst $i$ & $j$ free [c.f. Einstein summation convention].

vi) $T^{00}$ is just the energy density. For example if you have two particles orbiting each other, with positions $\mathbf{r}_1(t)$ and $\mathbf{r}_2(t)$ and masses $m_1$ and $m_2$, then you could write $T^{00}(\mathbf{r},t) = m_1 \delta^{(3)}(\mathbf{r} - \mathbf{r}_1(t)) + m_2 \delta^{(3)}(\mathbf{r} - \mathbf{r}_2(t))$.

[N.B. $\delta^{(3)}(\mathbf{r} - \mathbf{u}) \equiv \delta(x^1 - u^1) \delta(x^2 - u^2) \delta(x^3 - u^3)$].

 

[pervect]

MTW has an approximation of the formula given above in a post by ergosphere, which is known as the quadrupole formula. See page 978, section $36.2, "Power radiated in terms of internal p ower flow".

In non-geometric units, MTW's formula, which is derived in the context of one body orbiting another is:

$$P_{GW} \approx \frac{ \left( P_{internal} \right) ^2}{P_0} $$

$P_{GW}$ is the power (energy/unit time) radiated away by gravitational waves. It's a weak field approximation, so among other assumptions we are assumed there is no significant gravitational time dilation. The formula will work approximately for things like the Hulse-Taylor binary, it won't apply (nor will the quadrupole formula apply) in the strong-field regime of a pair of inspiraling black holes.

$P_0$ is a constant, equal to c^5/G, G being the gravitational constant. Numerically, in SI units it's $\approx 3.62 \, 10^{52}$ watts.

The large size of $P_0$ means that the ratio of the $P_{internal}$ to this constant in the weak field regime is small, much less than one.

$P_{internal}$ is the internal power flow of the system. It's described as the product of:

The mass of the part of the system that moves (in a circular orbit)​

( Size of the system^2 )​

( The reciprocal of the period of the system )^3​

The argument uses Kepler's law (this is a weak field approximation!), which in this context is:

The ratio of the square of an object's orbital period with the cube of the semi-major axis of its orbit is the same for all objects orbiting the same primary.

That's why it involves the ratio of the square of the radius of the orbit and the cube of the orbital period. $P_{internal}$ is equivalent to the non-spherical part of the energy of the system, $\approx m v^2$ per unit time, i.e. the non-spherical part of the power of the system.

The units work out, the square of a power divided by a constant power is a power.

This is an approximation of an approximation. There are various constant factors omitted.

If you still want to understand the quadrupole formulation better, the $I_{jk}$ is rather similar to the moment of inertia tensor, if you are familiar with the moment of inertia tensor. Then some diagonal elements are subtracted to make the trace of $I_{jk}$ vanish, making it the so-called "reduced" quadrupole tensor. Finally, to use the quardrupole formula, we need to take the third time derivative of this tensor. And to get the total emitted power, we'd have to project the resulting tensor expression to various angles from the source, then integrate. The complexities make it a bit hard to understand the physics, the formulation in terms of internal power flow makes the physics a bit more understandable. However, if you are interested in cases other than GW's emitted by a circular orbit, it's unclear to me how well MTW's formula would apply.