# Maximum power (luminosity) limit in General Relativity

Is the following thought experiment - to determine the maximum limit for power $$P_m$$ in General Relativity - correct?

It is by Craig J Hogan, a member of the team that discovered dark energy, so pretty mainstream I would think.

Consider an (almost) Kugelblitz sphere of radius $$R$$ $$\geqslant2Gp \over c^3$$ filled with photons with a total mass-energy of $$E=pc$$ that leave after a time $$t=R/c$$. Average power (luminosity) $$P=E/t$$. Therefore:

$$P_m$$ $$\leqslantc^5 \over2G$$

Original paper

The term $$c^5 \over G$$ is Planck power

• The assumption that the photons are spread out over a time $t$ could be violated. (E.g. they could all move outward on a single shell) Mar 13 '20 at 9:45
• @mmeent Thank you. Your comment inspired a search last week and I found a paper by Cardoso et al (2018) on this exact topic. I think I can answer the question myself now (i.e. yes there is a max power limit conjecture in GR) and will post it later if no solid refutations turn up. Mar 21 '20 at 6:23

There is a discussion of this in Misner, Thorne, and Wheeler, p. 980. Their treatment is more careful than Hogan's and more clear about the underlying assumptions. They derive this estimate by considering violent processes of astrophysical collapse in which the virial theorem is valid. They don't claim that it's an upper limit on the luminosity of any object.

Hogan's discussion seems to have a lot of logical problems. He starts by assuming a sphere "filled with light...and released [in] an instant." This is a very specific system, so there is no logical reason to conclude, as he does, that therefore, "An absolute luminosity limit for anything is imposed by General Relativity."

That would be pretty illogical, since, for example, if you had two such sources side by side, they would release twice the power, and you could consider them as one big object. I suppose MTW evade this counterexample because the two objects would not be equilibrated with one another, and therefore the virial theorem wouldn't apply to them if they were considered as a single system. I think this is also the answer to mmeent's counterexample of a thin shell of energy being released. It is indeed a counterexample to Hogan's claim, but not to MTW's, since the thin shell probably doesn't obey the virial theorem.

It's also weird and unfortunate that Hogan wants to connect this to the Planck mass. This would suggest, incorrectly, that there is something quantum-mechanical about this, when in fact (as he admits) there isn't. As he explains, the h-bars go away when you form units of power in Planck units.

• Interesting. I think the 'two sources side by side' is easily addressed by the fact that would create a black hole. You can see this in my answer below, as per Cardoso el al. As to the Planck mass, the speculative connection is obviously if you assign R=Planck length. May 24 '20 at 3:35
• Furthermore, Misner, Thorne, and Wheeler agree that maximum power occurs when then system is near its gravitational radius, although their max luminosity is the Planck power, not half this value as per Hogan or Cardoso. Jun 15 at 6:31

A: Yes. The existence of black hole horizons implies a maximum power limit in General Relativity, leading to related conjectures first posed around the time of Hogan’s 1999 paper. Figure 1 Outgoing shell of radiation and its past appearance

Consider a process (Figure 1, Cardoso et al 2018) with a finite time $$t$$ which produced radiation with a total mass-energy of $$E=Mc^2$$ contained in a shell of thickness $$R=c.t=1$$. Evolve this system into the past until $$R=shell radius$$. For the radiation to have escaped its initial self-gravity a BH horizon cannot have been present for the ‘particle’, so:

$$R ≥ 2GM/c^2$$ (1)

The maximum power, or luminosity, is therefore: $$P(max)=c.F(max)= E/t=c^5/2G$$ (2)

With natural units, we could assign $$R$$ as unit (Planck) length, $$t$$ is a unit (Planck) time so then $$E=1/2=pc=Mc^2$$ half a Planck mass-energy. The products (not the individual values), of these quantities are therefore minimum bounds in GR and quantum gravity, i.e. the uncertainty principle.

So $$MR=1/2$$ where $$R=ℏ/2Mc=1$$ being the reduced Compton radius of the ‘particle’. This was noted by Carter(1968), his Kerr-Newman solution for a charged and rotating black hole with a gyromagnetic ratio of 2 like the Dirac electron. That is, (2) also holds if the shell was charged and rotating.

(2) corresponds with the Planck acceleration $$a_P$$ being the maximum acceleration in quantum gravity.

$$F(max)= c^4/2G=M.a_P$$ (3)

(3) also agrees with Caianiello $$a_c=(2Mc^3)/ℏ=a_P$$

Maximum power supports the conjecture of maximum force (Gibbons) in GR, and therefore, maximum acceleration and temperature. (2) is twice the limit $$F_g$$ proposed by Gibbons, but of course, the local proper acceleration limit at $$R$$ is half $$a_P$$, which will also be the Hagedorn temperature in natural units, as noted by Lawrence B. Crowell.

To see this, consider the inverse square law and Figure 1. Gravitational intensity to a distant observer $$I=GM/R^2 =g=1/2$$ at the limit radius $$R=1$$. $$g$$ is acceleration, i.e. maximum gravitational field is $$κ=1/4M= 1/2=g$$. Then, from Unruh and Bekenstein:

$$T= (ℏ κ)/(2πck_B )=1/4π$$ (4)

$$S≤ (k_B c^3 πR^2)/ℏG=π$$ (5)

From (4) and (5) we can derive another limit, as per Padmanabhan

$$E=2TS$$ (6)

After all, the Gibbons GR force limit conjecture was for a force acting between two bodies. For example, the gravitational coupling constant, the gravitational attraction between two minimal mass-energies as above is the Gibbons (and Newtonian) limit.

$$α_G=(M/m_p )^2=G (M.M)/R^2 =1/4$$ (7)

The Gibbons force limit implies a local momentum flow limit, i.e. minimum mass-energy times maximum local proper acceleration $$F_g=M.g$$.

It also results from the entropic force $$F_g R=TS$$