Black holes and entropy: energy with respect to infinity

[cesiumfrog]

G'day!

In the paper "Black holes and entropy" (JD Bekenstein, Phys Rev D 7 2333, 1973), in the section on Geroch's perpetual motion* machine, I'm trying to understand why they can state "its energy as measured from infinity vanishes"?

What they mean is that the work extracted by lowering a test mass ($m$), "from infinity" (on an ideal string), to the (precise) surface of a black hole event horizon, is exactly $mc^2$.

I'm certainly not disagreeing: in Newtonian gravity, you can obtain infinite work by lowering one point mass toward another, but the point where the work (or - gravitational potential) equals the mass-energy happens to occur at half the Schwarzschild radius, where the approximation is invalid. Intuitively, it seems reasonable as, due to time dilation, the gravitational force becomes (permanently) zero such that it is a conceivable point where "all" of the work has been extracted. Can anyone give a more concrete explanation?

*The proposed machine takes a box "of black-body radiation" from a warm bath, lowers it toward a black hole elsewhere, and allows some energy to escape into the event horizon. The box is retrieved so as to repeat the process of converting heat into work with 100% efficiency, a violation of thermodynamics. The machine fails if the box can't quite be lowered all the way onto the surface (limited say by the radiation wavelength) so the efficiency is slightly less, and in fact turns out less than the Carnot efficiency given the black hole "temperature" (defined analogously with that in classical statistical mechanics, based on the similarity of thermodynamic entropy to event horizon surface area). It's an interesting paper, I'm reading it as a lead-up to Hawking radiation.

 

[pervect]

Just look at the defintion of "energy at infinity":

MTW pg 656 or online see http://www.fourmilab.ch/gravitation/orbits/

or consider that Energy-at-infinity = p_0, where p^0 is the energy-momentum 4-vector.

 

[Entropia]

Hi there! It's great to see that you're exploring the fascinating topic of black holes and entropy. Let me see if I can help clarify the concept of energy with respect to infinity in relation to black holes.

First, it's important to understand that in general relativity, energy is not a well-defined concept. This means that there is no universal way to measure the energy of a system, and it can vary depending on the observer's frame of reference. In the case of black holes, this becomes even more complex due to the extreme curvature of spacetime near the event horizon.

In the context of black holes and entropy, the concept of energy with respect to infinity refers to the energy of a system (in this case, the black hole) as measured by an observer at an infinite distance away. This is often used as a reference point because it is the farthest possible distance from the black hole, where the effects of its gravity are negligible.

In the specific example you mentioned from Bekenstein's paper, the work extracted by lowering a test mass from infinity to the event horizon is equal to mc^2. This is because at the event horizon, the gravitational force becomes zero and the work done to overcome it is equal to the mass-energy of the test mass.

As for the perpetual motion machine proposed by Geroch, it is important to note that it is a thought experiment and not a physically realizable device. The concept of energy with respect to infinity is used to show that the machine would violate the laws of thermodynamics, as it would be able to extract an infinite amount of energy without any input. However, as you mentioned, the efficiency of the machine would decrease as it approaches the event horizon, ultimately making it impossible to achieve 100% efficiency.

I hope this helps clarify the concept of energy with respect to infinity in relation to black holes. It's a complex and fascinating topic, and I encourage you to keep exploring it! Best of luck with your studies.