Force between black holes

[Martin]

here is a simplified calculation of the force between two black holes

that someone may have already done at SFN----it may not be new to you but it has an interesting result that connects to Christoph Schiller's

"fun with physics" thread

 

Two equal mass BH of the simplest kind (non-rotating electrically neutral), say each with mass M

 

the schwarzschild radius of each is

[math]R = \frac{2GM}{c^2}[/math]

 

so naively the closest they could get together is

 

[math]D = 2R = \frac{4GM}{c^2}[/math]

 

and then by newton's law of gravitation the force of attraction between

them is

 

[math]F = \frac{GM^2}{D^2}[/math]

 

but

 

[math]D^2 = \frac{16G^2 M^2}{c^4}[/math]

 

so the M cancels out of the formula for the force and one gets

 

[math]F = \frac{c^4}{16G}[/math]

 

this is 1/16 of a kind of benchmark force sometimes called the Planck force because it is the unit of force corresponding to planck unit mass and unit length and unit time etc. what you call it doesnt matter. it is

 

[math]F_p = \frac{c^4}{G}[/math]

 

and it is around 10⁴⁰ tons

 

I havent looked at the individual pages of Christoph's book but i took a look at his TOC just now and I see that he proves that the strongest possible force in nature is a quarter of this.

 

that is interesting and makes sense

if the two black holes were approaching each other at relativistic speed they could get closer than I said and experience a stronger force of attraction (my calculation is not the best one can do, so one should be able to get it to be better than 1/16 and maybe 1/4 is right)

 

but aside from factors like 1/4 and 1/16, here we have a handle on

the strongest force of attraction between two distinct objects---the instant right before they merge into one.

[timo]

Well, maybe you could tell us what you want to say with this post?

 

- Do you want to know if your calculation is correct? Well, black holes only appear in general relativity as a coordinate singularity. As I allready said in another post there is no forces due to gravity in GR. Calculating celestial problems with Newtonian Physics (forces) is ok as long as gravitational fields are not too strong. But the Schwarzschild-radius you picked is one of the points where Newtonian Mechanics ultiamtely fails. That´s why it became so popular.

 

- Did you want to show something with that calculation? For example what is the greatest force due to gravity that can exist? The term "strongest possible force" is almost certainly incorrect. You could locate two charges at any distance you want exceeding any force if the distance is small enough. Also "strongest gravitational force" would not be correct because in your picture nothing would prevent you to add further black holes in a way that the two forces add up to a bigger force (triangle).

 

- One thing I didn´t understand but what would really interest me. Given two equal black holes at a distance of twice their Schwarzschild-Radius:

What makes you think they´ll merge into one at this point? Why not at smaller or greater distances? Why at all?

-> Is that what you intuitively think? Yes, it would be intuitively clear. But things like the Schwarzschild-radius are not intuitively understandable anyways.

-> Did you read that somewhere in the internet? If yes, what did the original poster say about that?

-> Do you have a scientific proof for that? I really don´t know if someone did a relativistic calculation of such a system, but I´d be very interested if there´s one. On the one hand there is no known general solution of the Einstein-equations but on the other hand there are many symmetries that can be used so there might be one.

[Martin]

I want to call attention to motionmountain's "fun with physics" thread in this forum. his name is Christoph Schiller and he has a 1000page online physics fun-book. One thing christoph has is a chapter on things like the "strongest possible force" in nature.

 

Basically the idea is, we all know that in a certain sense c is a natural speed limit. It is the fastest possible speed. (if you define terms properly)

 

so is there an upperbound on the (rest)mass of a particle or is there an upperbound on the power any system can deliver or on force?

or is there a smallest length that can be distinguished by measurement etc etc.

 

these questions are kind of in the commonplace zeitgeist vernacular----well Christoph has some words about them

 

I am very pleased with your response, Atheist. thanks for replying.

 

You ask about my own history with these ideas. years back when I first heard of planck time, length, mass it occurred to me to calculate what the force unit is that goes with those natural units and I calculated that it was c⁴/G and I immediately asked myself if that force, which is E40 tons, has some extremal property

after all c is the planck unit of speed---one planck length per planck unit of time----and several planck units do seem to be extremal (smallest measureable distance etc)

 

so I did a little back of envelope calculation with black holes. At that time I put the singularity of one on the event horizon of the other and got IIRC the answer of

c⁴/4G

 

I dont think order-of-unity factors matter in such a rough conceptual calculation, so I dont think the exact answer is important. Only that if two BH are falling towards one another then as long as they are separate entities one can assign a momentum to each one

and one can in principle observe their acceleration

and from the change in momentum one can define a force

(a rate of change of momentum)

so it is interesting to know what is the maximum rate of change of momentum they can have up to the point that one can no longer

distinguish them as separate entities

 

this is not a mathematically rigorous thing---what does separate mean?---but it is a nice backofenvelope calculation that gives an idea of natures extreme force

 

I think your idea of putting two electrons close together to feel their extreme repulsion is also interesting.

But I do not think you can achieve E40 tons of force this way because

of things like the HUP and the compton wavelength of the electron---or whatever particle you are imagining.

 

when they get too close they are no longer distinguishable or they simply cannot exist that close together---something breaks.

 

so i think you will find that you get a bigger force if you do it my way.

 

now what Christoph does, I dont know! I didnt read his book yet. his chapters take a while to download. maybe someone who has read the Schiller book can tell us.

 

I also am not sure your "triangle" idea works to give a larger force.

whatever mass you have it may be most effective to concentrate it in just two BHs-----that may be the setup that gives the highest rate of change of momentum

 

Glad to see you trying these ideas out! Maybe you will come up with

a physical system in which there is a stronger force than what

Christoph says namely c^4/4G

 

Well' date=' maybe you could tell us what you want to say with this post?

 

- Do you want to know if your calculation is correct? Well, black holes only appear in general relativity as a coordinate singularity. As I allready said in another post there is no forces due to gravity in GR. Calculating celestial problems with Newtonian Physics (forces) is ok as long as gravitational fields are not too strong. But the Schwarzschild-radius you picked is one of the points where Newtonian Mechanics ultiamtely fails. That´s why it became so popular.

 

- Did you want to show something with that calculation? For example what is the greatest force due to gravity that can exist? The term "strongest possible force" is almost certainly incorrect. You could locate two charges at any distance you want exceeding any force if the distance is small enough. Also "strongest gravitational force" would not be correct because in your picture nothing would prevent you to add further black holes in a way that the two forces add up to a bigger force (triangle).

 

- One thing I didn´t understand but what would really interest me. Given two equal black holes at a distance of twice their Schwarzschild-Radius:

What makes you think they´ll merge into one at this point? Why not at smaller or greater distances? Why at all?

-> Is that what you intuitively think? Yes, it would be intuitively clear. But things like the Schwarzschild-radius are not intuitively understandable anyways.

-> Did you read that somewhere in the internet? If yes, what did the original poster say about that?

-> Do you have a scientific proof for that? I really don´t know if someone did a relativistic calculation of such a system, but I´d be very interested if there´s one. On the one hand there is no known general solution of the Einstein-equations but on the other hand there are many symmetries that can be used so there might be one.[/quote']