Gravitational energy between two point masses infinite? Surely not

[bcrelling]

If you draw a graph representing the tapering of gravitational force with respect to distance between two point masses (by the inverse square law y=x<exp-2>), the gravitational energy between two points would be the area under the graph between those points. This is my assumption.

Now the area is calculated by integrating between those points.
And the area turns out to be infinite!

Any ideas on what's going on- what am I missing here?

 

[ZapperZ]

If you draw a graph representing the tapering of gravitational force with respect to distance between two point masses (by the inverse square law y=x<exp-2>), the gravitational energy between two points would be the area under the graph between those points. This is my assumption.

Now the area is calculated by integrating between those points.
And the area turns out to be infinite!

Any ideas on what's going on- what am I missing here?

When integrating a function that has poles, you cannot simply integrate "normally". You have to invoke mathematical techniques such as Residue theorem, etc.

Zz.

 

[jbriggs444]

You are not missing anything. Your observation is correct. In the Newtonian model, arbitrarily large quantities of energy can be extracted if two point masses are allowed to approach each other sufficiently closely.

This is a clue suggesting that either the Newtonian model is incomplete or that pointlike masses are impossible. As we know, the Newtonian model is incomplete. General relativity extends/replaces it. The uncertainty principle in quantum mechanics throws a bit of a monkey wrench into the ordinary notion of "pointlike".

 

[Nugatory]

If you draw a graph representing the tapering of gravitational force with respect to distance between two point masses (by the inverse square law y=x<exp-2>), the gravitational energy between two points would be the area under the graph between those points. This is my assumption.

Now the area is calculated by integrating between those points.
And the area turns out to be infinite!

Any ideas on what's going on- what am I missing here?

That happens because the force goes to infinity under the inverse square law as the distance goes to zero near a point source; it really would take an infinite amount of energy to separate two point sources starting at the same point. But in real life there are no point sources of gravity; every object has some non-zero size. For example, if you're working with the gravitational force of the earth, r will never be less than 4000 miles - any less than that and the surface of the Earth would get in the way. You need to allow for this when choosing the range for your integration.

 

[bcrelling]

Ok, thanks guys.

Any suggestions on what the closest two masses can get to each other(in order to calculate the maximum extractible gravitational energy) in absolute terms?

A plack length or distance between two quarks perhaps?

 

[HallsofIvy]

It makes no sense to talk about "point masses" while talking about "planck length" and "quarks". The first is a mathematical abstraction while the others are real physical objects.

In any case you get the "maximum extractible energy" by making two bodies far apart, not close together.

 

[mrspeedybob]

Ok, thanks guys.

Any suggestions on what the closest two masses can get to each other(in order to calculate the maximum extractible gravitational energy) in absolute terms?

A plack length or distance between two quarks perhaps?

The closest thing to a point mass that actually exists would be a black hole so the greatest amount of gravitational energy release would be when 2 black holes merge. The mathematics of this scenario is fairly nasty, and a bit beyond me right now, but that's the direction you need to look for your answer.

 

[bcrelling]

The closest thing to a point mass that actually exists would be a black hole so the greatest amount of gravitational energy release would be when 2 black holes merge. The mathematics of this scenario is fairly nasty, and a bit beyond me right now, but that's the direction you need to look for your answer.

Thanks that makes a lot of sense, and we needn't consider only defacto black holes, as technically every particle has it's own Schwazschild radius. I'll see it I can integrate between the event horizon and infinity.

 

[bcrelling]

The closest thing to a point mass that actually exists would be a black hole so the greatest amount of gravitational energy release would be when 2 black holes merge. The mathematics of this scenario is fairly nasty, and a bit beyond me right now, but that's the direction you need to look for your answer.

I realized that calculating the energy between the event horizon and infinity is actually simpler and might not require integrating.
We know that the escape velocity at the event horizon is the speed of light- escape velocity by definition being the speed required to take a mass to an infinite distance from source.

So using a 1kg mass at speed C in the Newtonian formula for kinetic energy(MV²/2) would give us half the energy that same mass would contain at rest(from E=MC²).

However I assume that we'll have to use relativistic kinetic energy, and so the energy required to take a mass to escape velocity from the event horizon is going to be infinite. So now we're back where we started- considering that it takes a finite amount of energy to achieve escape velocity at any distance greater than the event horizon the gravitation energy of anything falling from any distance into a black hole is must be infinite.

That would suggest there is infinite gravitaional energy between any two masses allowed to fall into each others event horizons. Again, this can't be right?

 

[Crazymechanic]

I think a honest answer would be that we don't know., WHY?
Because we can't see , basically beyond the event horizon.WHY?
Because light doesn't reflect back , WHY?
Because after it reaches the EH the escape velocity gets greater than c so it can't get back even if it wanted to...


"""" However I assume that we'll have to use relativistic kinetic energy, and so the energy required to take a mass to escape velocity from the event horizon is going to be infinite.""""

According to theory this is right.
Now we know about black holes from the fact that EM radiation (visible light included) get's absorbed by a BH and everything around it greatly distorted so if you ask me I don't know what is inside of it but the fact that the force required to "suck in" mass at a greater than c speed is infinite seems legit.