The spectrum of gravitational waves!!!!! Purely gravitational wave music!
The blog post today will cover two topics from elementary viewpoints: falling into a non-rotating black hole and gravitational wave “music”, i.e., gravitational wave formulae! It is a hard equilibrium just to fall into the BH singularity and to be spaghettified, but you will be awarded with gravitational wave physics at the second act! Happy???
Non-rotating black holes are called Schwarzschild black holes, or Schwarzschild-Tangherlini, since the latter generalized the black hole metric into extra space-like dimensions in 1963. I will keep mathematics as simple as possible, but that will introduce some imprecisions that, I wish, experts in the field will forgive.
A classical particle owns a lagrangian
It implies that
Firstly, let us consider a test particle at rest from an infinite distance from the black hole event horizon, with , and thus set up
Note that the above result is essentially the classical escape velocity. Secondly, make a trip towards the singularity, starting from (at ). Supposing its distance from is you get
and now proceed to integrate the above equation with the proper above mentioned limits
Remark: until the singularity from the boundary event horizon radius, you get a distance , until the initial point, you have a distance , until the final point you have from the initial point but from the singularity at the center.
After integration, and reintroducing , you will obtain that
If and for the time we reach the singularity, then
This calculation can be performed for a -dimensional black hole with space-like dimensions. If you define the higher dimensional version of the Schwarzschild radius the analogue integration is about
A variation of this time (but you will be likely death before the singularity arrives), can be done if you use the same geodesic equation above but with initial (or a big test mass so ), then the integral is lightly different and you have to be careful to evaluate it. I will calculate it only in the usual spacetime to compare it with the previous result:
The result is
The difference is not big, since , so we can be sure that we will be pushed into the singularity in about that order of time. Of course, the caveats are the life of the passenger and that quantum gravity should be taken into account at some point in the interior of the black hole. Maybe even before, according to the firewall paradigm.
What is next? Small problem: what is the density for a spherical exoplanet, moon or compact object to stay weightlessness in the equator surface? For 3d space, we can equalize
and plugging and , , then we will obtain
By the other hand, our Universe is mysterious. Just like when we have different types of gravitational lensing (strong lensing, weak lensing and microlensing),via e.g. a simple equation
the study of the gravitational waves just began. Perhaps, we will have new tools to test even the black hole entropy of our almost de Sitter Universe. The dS Universe entropy is given by
For Keplerian orbits, the same argument than the previous one will help us to find the gravitational wave frequencies. The orbital frequency for quasi-circular keplerian orbits will be
The gravitational wave frequency in Einstein theory is twice the orbital frequency, i.e., , and then
or, in terms of density,
We want to compute this quantity from some reference scale, e.g., solar mass scale. Then, since and are the Schwarzschild radius and the solar mass respectively, by ratios we can calculate
Therefore, the dominant gravitational wave frequency for quasicircular keplerian orbits reads off as
In the end, unless quantum gravity changes the rules, any gravitational bounded system will decay gravitationally! The time for the gravitational coalescence of two orbiting bodies can also be computed from General Relativity for any (even eccentric) orbit. If initially the two bodies, with masses have a separation and eccentricity , then the time till coalescence will be
withe the eccentricity function
Since is the major semi-axis, the apoastron and periastron centers will be , . Also, define the chirp mass
Then, the time to coalescence is
while the peak of the gravitational wave frequency (music) will be
See, e.g., Wen 2003, or Antonini et al. 2014. to check the above formulae. By the other hand, merger anatomy is generally complex and hard with current tools. There are phases: inspiral, plunge, merger and coalescence. The final phase is “balding hair” of the black hole (in the case of binary black hole mergers). After plunge and merger phase end, there is also a new phase called “ring-down”. The final product is a Kerr black hole in general. At least from the current knowledge of gravity, general relativity and black hole theory. Kerr black hole is perturbed and the so-called quasinormal modes are emitted. For quadruple QNM (quasinormal mode), from the paper PRD 34, 384 (1986), you can get the values
where is the Kerr parameter for the final state. In this field, there are also zoom-whirls. Zoom-whirl orbits are perturbations of unstable circular orbits that exist within the inner stable circular orbits (ISCO). The number of whirls is related to perturbation magnitude and instability exponent via . Orbit taxonomy and classification of Kerr-like black hole orbits are also possible. It is generally defined a rational number , where is the number of whirls, and is the number of leaves that make up the zooms, while is the sequence in which the leaves are traced out (). Any non-closed orbit is arbitrarily close to some periodic orbit. If , then .
There are two more places where gravitational waves are important. Firstly, for the so-called Kozai-Lidov resonances in ternary systems. When a binary system is perturbed by a third body, the latter can induce some periodic oscillations in the libration of the orbital ellipse and a variation in the orbital eccentricity. The typical Kozai-Lidov oscillations can be calculated to have a period
where is the period of the triplet’s inner orbit (keplerian), is the period of the triple’s outer orbit (keplerian).
Finally, we have the so-called stochastic gravitational wave background, a buzz in GW caused by non-resolved GW sources. This is different from the so-called gravitational or graviton relic background (the cosmic analogue of the cosmic microwave background). Just as the current CMB has a temperature , and the cosmological neutrino background has a temperature , the relic graviton background is expected to be . For the stochastic background, however, we define the energy density
and the gravitational wave cosmological stochastic density to be
Therefore, the strain spectrum and the strain scale are respectively
Where do all of those equations come from? Consider a small perturbation of the spacetime metric
The Einstein’s Field Equations read off
In vacuum implies that can be used to compute
and plus sources
The simplest solution to a wave-equation is the plane wave
With many simplifications, the solution is rewritte as the variation of a quadrupole moment
There, the distance to the source is and the quadrupole moment is delayed or computed in the so-called retarded time. This is easier than the entropic uncertainty principle for sure. Gravitational waves from binaries, during the inspiral phase, is just elementary physics from these considerations. First, assum edge on observation () such as . Then, you can check that ( is the retarded time):
and then, with the previous conditions, deduce that
where , and with the aid of Kepler third law you get
with . Now, resume the formulae
Moreover, we can calculate the gravitational wave power radiated in the GW emission as
and the time varying angular frequency will be
and for quasicircular orbits, you get the previously mentioned value
while the general formula in the literature is
From Peter and Mathews, we also have
Integrating the semi-axis from to , you would get
For the major semi-axis shrinking value we get in the literature being similar
and the integration constant
The function of radial shrinking in terms of eccentricity reads
The period decreases as
The power decrease is given by the formula
where is, as before,
Maggiore’s book on gravitational waves define
See you in other blog post!
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