LOG#223. Pi-logy.

Posted on by July 19, 2020

Hi, there.

Today some retarded Pi-day celebration equations (there is a longer version of this, that I wish I could publish next year). Some numbers and estimates for pi-related equations:

1st. Hawking radiation temperature (Schwarzschild’s 4d black hole case).

(1)   \begin{equation*}T_H=\dfrac{\hbar c^3}{8\mathbf{\pi} G_NMk_B}=6.2\cdot 10^{-8}\left(\dfrac{M}{M_\odot}\right)K\end{equation*}

2nd. Schwarzschild black hole surface area (4d).

(2)   \begin{equation*}A_S=4\mathbf{\pi}R_S^2=\dfrac{16\mathbf{\pi}G_N^2M^2}{c^4}=1.1\cdot 10^8\left(\dfrac{M}{M_\odot}\right)^2\; m^2\end{equation*}

3rd. Black hole power/luminosity (4d).

(3)   \begin{equation*}L_{BH}=P_{BH}=\dfrac{\hbar c^6}{15360\mathbf{\pi}G_N^ 2M^2}=9.0\cdot 10^{-29}\left(\dfrac{M_\odot}{M}\right)^2W\end{equation*}

4th. Black hole evaporation time (4d).

(4)   \begin{equation*}t_{e}=\dfrac{5120\mathbf{\pi}G^2_NM_0^3}{\hbar c^4}=8.41\cdot 10^{-17}\left(\dfrac{M}{1kg}\right)^3s=6.6\cdot 10^{74}\left(\dfrac{M}{1kg}\right)^3s=2.1\cdot 10^{67}\left(\dfrac{M}{1kg}\right)^3yrs\end{equation*}

5th. Time to fall off and arrive to the BH singularity with negligible test mass (4d).

(5)   \begin{equation*}t_f(test)=\dfrac{\mathbf{\pi}}{2c}R_S=\dfrac{\mathbf{\pi}G_NM}{c^3}=1.5\cdot 10^{-5}\left(\dfrac{M}{M_\odot}\right)s\end{equation*}

6th. Time to fall off and arrive to the BH singularity with E=m test mass (4d).

(6)   \begin{equation*}t_f(m)=\dfrac{2}{3}\dfrac{R_S}{c}=\dfrac{4\mathbf{\pi}G_NM}{c^3}=6.2\cdot 10^{-5}\left(\dfrac{M}{M_\odot}\right)s\end{equation*}

7th. Black hole entropy (4d) value in SI units.

(7)   \begin{equation*}S=\dfrac{k_B c^3}{G_N\hbar}A_{BH}=\dfrac{k_BA}{4L_p^2}=\dfrac{4\mathbf{\pi} GM^2}{\hbar c}=\dfrac{\mathbf{\pi}k_Bc^3A_{BH}}{2G_Nh}=1.5\cdot 10^{54}\dfrac{M^2}{M_\odot^ 2}J/K\end{equation*}

8th. M2-M5 brane quantization.

(8)   \begin{equation*}T_{M2}T_{M5}=\dfrac{2\mathbf{\pi}N}{2k_{11}^2}=\dfrac{\mathbf{\pi}N}{k_{11}^2}\end{equation*}

9th. Gravitational wave power or GW luminosity.

    \[L_{GW}=-\dfrac{dE}{dt}=\left(\dfrac{32}{5c^5}\right)G^{7/3}\left(M_c\pi f_{GW}\right)^{10/3}\]

where the gravitational wave frequency is

    \[f_{GW}=2f_{orb}=\dfrac{1}{\mathbf{\pi}}\sqrt{\dfrac{GM}{r}}\]

10th. Chirp frequency or frequency rate.

For circular orbits, you have

    \[\dot{f}_{GW}=\left(\dfrac{96}{5c^5}\right)G^{5/3}\pi^{8/3}M_c^{5/3}f_{GW}^{11/3}\]

11th. Coalescence time for GW merger (circular orbits).

    \[t_c=\dfrac{1}{2^8}\left(\dfrac{GM_c}{c^3}\right)^{-5/3}\left[\mathbf{\pi}f_{GW}\right]^{-8/3}\]

12th. ISCO (inner stable circular orbit) frequency for binary mergers.

    \[f_{max,c}=f_{isco}=\dfrac{c^3}{6^{3/2}\pi GM}\approx 4.4\dfrac{M}{M_\odot} kHz\]

13th. S-matrix in D-dimensions.

    \[S=I+i\dfrac{\left(2\pi\right)^D\delta^D\left(\displaystyle{\sum_fp_f}-\displaystyle{\sum_ip_i}\right)}{\displaystyle{\prod_f}\left(2p_{of}\right)^{1/2}\displaystyle{\prod_i}\left(2p_{oi}\right)^{1/2}}\mathcal{A}\]

14th. Gravitational wave fluxes for gravitons and photons (4d).

    \[F_{GW}=\dfrac{c^3h^2\omega^2}{16\pi G_N}=\dfrac{\pi c^3h^2f^2}{4G_N}\]

where h is the GW strain, and for photons, the GW induced electromagnetic  flux reads

    \[F_{em}=\dfrac{c^3\omega^2 h^4}{8\pi G_N}=\dfrac{\pi c^3 f^2 h^4}{2G_N}\]

15th. Kerr-Newmann black hole area and mass spectrum.

Any massive, rotating, charged black hole have an event horizon given by the following formula

    \[\mathcal{A}_H=4\pi\left[\dfrac{2G_N^2M^2}{c^4}-\dfrac{G_NQ^2}{4\pi\varepsilon_0c^4}+\dfrac{2G_NM}{c^2}\sqrt{\dfrac{G^2_NM^2}{c^4}-\dfrac{G_NQ^2}{4\pi\varepsilon_0c^4}-\dfrac{J^2}{M^2c^2}}\right]\]

This relation can be inverted to obtain the mass spectrum as function of area, charge and angular momentum as follows (exercise!):

    \[\mathcal{M}\left(A_H,Q,J\right)=\sqrt{\dfrac{\pi}{\mathcal{A}}}\left[\dfrac{c^4}{G_N}\left(\dfrac{\mathcal{A}}{4\pi}+\dfrac{G_NQ^2}{4\pi\varepsilon_0c^4}\right)^2+\dfrac{4J^2}{c^2}\right]^{1/2}\]

Challenge: modify the above expressions to include a cosmological constant factor.

16th. Universal quantum gravity potential at low energies.

Quantum gravity at low energy provides the following potential energy

    \[V_{QG}=-\dfrac{GM_1M_2}{r}\left[1+\dfrac{3G_N\left(M_1+M_2\right)}{rc^2}+\dfrac{41G_N\hbar}{10\pi r^2}\right]\]

independent of the QG approach you use!

17th. Running alpha strong.

    \[\alpha_s(Q^2)=\dfrac{\alpha_s(\Lambda^2_{QCD})}{1+\beta\alpha_s(\Lambda^2_{QCD})\log\left(\dfrac{Q^2}{\Lambda^2_{QCD}}\right)}\]

For the general QCD the beta factor reads

    \[\beta=\dfrac{11N_c-2n_f}{12\pi}\]

and the SM gives \beta_0>0 (N_c=3, n_f=6) and slope \beta(\alpha_s)<0 due to asymptotic freedom (antiscreening).

18th. Graviton energy density and single graviton energy density.

The graviton energy density reads off from GR as

    \[\rho_E=\dfrac{c^2\omega^2f^2}{32\pi G_N}\]

and for a single graviton, it reads

    \[\rho_E(single)=\dfrac{\hbar \omega^4}{c^3}=\dfrac{8\pi^3 h f^4}{c^3}\]

where h is the Planck constant, not the strain here.

I have many other pi-logy equations, but let me reserve them for a future longer post!

See you all, very soon!

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