# Against or for maximum force

• A
JohannesSK
The number of papers on maximum force c^4/4G and similar limits in general relativity is growing. Last year there were
A. Jowsey et al, arXiv:2102.01831​
V. Faraoni, Phys Rev D 103, 124010 (2021)​
C. Schiller, Phys Rev D 104, 068501 (2021)​
V. Faraoni, Phys Rev D 104, 068502 (2021)​
C. Schiller, Phys Rev D 104, 124079 (2021)​
L. Cao et al, Phys Rev D 104, 124017 (2021)​
Of these papers, 2 are against, 3 are in favour, 1 is open/changed his mind. It's a controversial topic.
Either a limit force exists, or it does not.
What are the pros and cons that count?
Is there a kind of argument that settles the question?

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Since top experts are arguing among themselves within the last year, it sure seems the question has no clear consensus yet. Thanks for this post, I had not seen the maximum force conjecture before seeing your post and scanning a couple of these papers.

I found free links to 4 of the references. The second Faraoni one appears to have no free copy available, and the Cao paper appears irrelevant (https://arxiv.org/abs/2109.05973).

https://arxiv.org/abs/2102.01831
https://arxiv.org/abs/2112.15418
https://arxiv.org/abs/2109.07700
https://arxiv.org/abs/2105.07929

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Nugatory
One thought that occurs to me is that essentially every GR phenomenon has some analog in SR (e.g. expansion, cosmological redshift, superluminal recession[Milne 'universe], invariant c, horizons [Rindler], energy conditions, etc.). What would be the possible SR basis for a maximum force? At first glance, this makes me skeptical of the hypothesis, especially as Schiller claims it is a local limit.

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Mentor
What would be the possible SR basis for a maximum force?
I don't think there is one. I don't really think the maximum force idea is a "GR" idea either. I think the proposal is best viewed as similar to proposals of standard GR breaking down at the Planck length, even though, as the paper by Jowsey and Visser that was referenced remarks, Planck's constant does not appear in the maximum force formula. They also remark that the "maximum force" value appears in the Einstein Field Equation. I believe that, according to the EFE, a force being equal to the conjectured "maximum force" would correspond to an energy density of whatever was producing the force being of the same order as the Planck density.

geshel and vanhees71
On a quick search, I couldn't find any exploration in the literature of this idea, but I wonder about the EM force between 2 superextremal Reissner-Nordstrom BH (with charge > mass, and no horizon, i.e. naked singularity). It would seem to me, that there would be no bound on EM force in this case, between such objects of opposite charge. Schiller does a lot of arguing based on horizons, which would not apply in this case.

vanhees71
Staff Emeritus
The whole concept of maximum force seems fishy.

Suppose there is a maximum force. Now have two anvils each pushing with 3/4 the maximum force. No problem, right? Now attach them with a loose string. We just exceeded the limit. Huh? How does it know?

vanhees71
JohannesSK
One thought that occurs to me is that essentially every GR phenomenon has some analog in SR (e.g. expansion, cosmological redshift, superluminal recession[Milne 'universe], invariant c, horizons [Rindler], energy conditions, etc.). What would be the possible SR basis for a maximum force? At first glance, this makes me skeptical of the hypothesis, especially as Schiller claims it is a local limit.
If the force is given by c^4/4G, it seems to me that it cannot have a SR basis, because G does not arise in SR.

vanhees71
JohannesSK
Suppose there is a maximum force. Now have two anvils each pushing with 3/4 the maximum force. No problem, right? Now attach them with a loose string. We just exceeded the limit. Huh? How does it know?
If I read correctly, adding forces at different locations does not count as exceeding the limit.

vanhees71
JohannesSK
On a quick search, I couldn't find any exploration in the literature of this idea, but I wonder about the EM force between 2 superextremal Reissner-Nordstrom BH (with charge > mass, and no horizon, i.e. naked singularity). It would seem to me, that there would be no bound on EM force in this case, between such objects of opposite charge. Schiller does a lot of arguing based on horizons, which would not apply in this case.
Do naked singularities exist?

JohannesSK
At least this, by G Gibbons: https://arxiv.org/abs/hep-th/0210109
Makes no mention of Reissner-Nordstrom BH. Also, argues based on on horizons, when my whole point is that horizons are not a requirement of GR as a classical, mathematical physics theory.

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If the force is given by c^4/4G, it seems to me that it cannot have a SR basis, because G does not arise in SR.
That's not relevant for an SR analog. Gravitational redshift has a perferct SR analog, even though the constant G has no meaning in SR. They key point is that the mathematical structure of GR requires it to be locally SR, so how can there be a local GR phenomenon without an SR analog?

vanhees71
Do naked singularities exist?
If the argument is about what plausibly exists, then this hypothesis is more like Hawking's "cosmic censorship conjecture" rather than a more fundamental conjecture. The point is that superextremal Reissner-Nordtrom BH certainly exist as part of GR as a classical, mathematical physics theory. Similarly, closed timelike curves exist in the mathematical theory. Then, I would have expected to see more substantive discussion of what must be assumed to make the conjecture true e.g. dominant energy condition, or existence of a Cauchy surface with evolution from initial conditions not of measure zero (Hawking's later formulation of his censorship conjecture after the simple one was disproved).

If I read correctly, adding forces at different locations does not count as exceeding the limit.
The string can be made arbitrarity short.

Mentor
I wonder about the EM force between 2 superextremal Reissner-Nordstrom BH (with charge > mass, and no horizon, i.e. naked singularity).
The EFE is nonlinear, so you can't superpose two RN solutions this way. Nor can you make use of any weak field approximation to say that the nonlinearity will be negligible.

vanhees71
The EFE is nonlinear, so you can't superpose two RN solutions this way. Nor can you make use of any weak field approximation to say that the nonlinearity will be negligible.
You can’t superpose them linearly, but you certainly can posit two of them moving radially towards each other. I didn’t state any firm conclusion, but merely asked whether the case was considered. My intuition is certainly that EM force would grow without bound, but I never claimed this as fact.

JohannesSK
Makes no mention of Reissner-Nordstrom BH. Also, argues based on on horizons, when my whole point is that horizons are not a requirement of GR as a classical, mathematical physics theory.
What does "not a requirement" mean? GR does have horizons.

What does "not a requirement" mean? GR does have horizons.
To the extent the maximum force conjecture relies on horizons (several of the arguments in the papers rely on horizons), then:

a) the conjecture is not local, contrary to author's claims. Horizons are inherently non-local features.
b) it cannot be a general feature of GR because GR allows Reissner-Nordstrom BH without horizons that can get close enough to have unbounded EM force between them.

JohannesSK
The string can be made arbitrarity short.
Surely, but not zero. So this does not seem a counterexample.

Surely, but not zero. So this does not seem a counterexample.
Once two force vectors are close enough that any parallel transport of one to the other cannot change them more than a tiny amount, they can be added as ordinary vectors (delta a tiny amount). This also touches on my argument that for this conjecture to a fully general statement about classical GR, it must have an analog in SR, because in a sufficiently small spacetime region, no physical distinction exists between SR and GR.

If they were saying this law relates to non-local features, and relates to things like BH horizons, the above argument would fail. But to the extent it is claimed to be universal and local, the above argument disproves the conjecture.

JohannesSK
The number of papers on maximum force c^4/4G and similar limits in general relativity is growing. Last year there were
A. Jowsey et al, arXiv:2102.01831​
V. Faraoni, Phys Rev D 103, 124010 (2021)​
C. Schiller, Phys Rev D 104, 068501 (2021)​
V. Faraoni, Phys Rev D 104, 068502 (2021)​
C. Schiller, Phys Rev D 104, 124079 (2021)​
L. Cao et al, Phys Rev D 104, 124017 (2021)​
Of these papers, 2 are against, 3 are in favour, 1 is open/changed his mind. It's a controversial topic.
Either a limit force exists, or it does not.
What are the pros and cons that count?
Is there a kind of argument that settles the question?

They also talk about the maximum power c^5/4G and the maximum mass flow rate c^3/4G.

I wonder whether a maximum mass flow rate limits the speed of a Lamborghini. A Lamborghini is about L=4 m long, weights M=1000 kg. With a maximum mass flow rate its resulting maximum speed is V = c^3/4G * L/M. Too bad, the limit is much larger than c.

Maybe a lower limit is found if gamma is inserted?

They also talk about the maximum power c^5/4G and the maximum mass flow rate c^3/4G.

I wonder whether a maximum mass flow rate limits the speed of a Lamborghini. A Lamborghini is about L=4 m long, weights M=1000 kg. With a maximum mass flow rate its resulting maximum speed is V = c^3/4G * L/M. Too bad, the limit is much larger than c. Maybe gamma must be inserted?
I think these limits and their arguments are quite different. For example, the Cao paper addresses a supposedly related limit on luminosity. I find the Cao result convincing for what it actually claims - the Bondi mass decrease rate of a radiating body is limited, which limits the luminosity. But Bondi mass is strictly global measure (must be done over the whole spacetime), and is defined only for asymptotically flat spacetimes. As such, it appears to have no bearing on a claimed local maximum force conjecture. Further, the Cao argument doesn't apply at all to our universe, because all FLRW spacetimes (except one degenerate case - no matter, energy, or cosmological constant) are not asymptotically flat.

JohannesSK
Once two force vectors are close enough that any parallel transport of one to the other cannot change them more than a tiny amount, they can be added as ordinary vectors (delta a tiny amount). This also touches on my argument that for this conjecture to a fully general statement about classical GR, it must have an analog in SR, because in a sufficiently small spacetime region, no physical distinction exists between SR and GR.

If they were saying this law relates to non-local features, and relates to things like BH horizons, the above argument would fail. But to the extent it is claimed to be universal and local, the above argument disproves the conjecture.
This is a good point! Is a horizon a non-local feature?

This is a good point! Is a horizon a non-local feature?
Yes, a an event horizon is inherently a global feature. The related notion of trapped surface is "quasi-local", but I don't recall trapped surfaces being mentioned in Schiller's papers (and quasi-local is still not local).

JohannesSK
The string can be made arbitrarity short.
I saw that strings are discussed in one paper. It is seems that pulling a string cannot exceed maximum force.

Staff Emeritus
This is possibly irrelevant, but Misner, Thorne, Wheeler (MTW) has a section in "Gravitation" on maximum power in general relativity, the details of which I don't recall. It may have been restricted to the weak field approximation. Anyway, power, force, and velocity are all dimensionless in geometric units. And on dimensional grounds, if there is a maximum power, one could divide it by c to get a maximum force, though I'm not sure that would have much physical significance. I'm not very familiar with the topic and haven't read any of the refrences, but I thought the MTW section on maximum power might be relevant.