Ultimate size of the Universe (finite)

[AJustice]

Ok, I am a mechanical guy, mechanical engineering degree. With an argument with my Engineering physics wife.

1. Imagine 2 particles (lightest particle with any finite mass)
2. separated by the the entire distance of the universe
3. the particles are attracted (due to gravity) to each other and will require an infinite amount of energy to reach full "light speed"

The particles will accelerate (increase mass due to relativity) as they move toward another and speed up. This will require an infinite amount of energy.

So, I say that if we assume the entire Universe is reduced to two "lightest" particles, the Universe MUST have a MAXIMUM size because two of the lightest particles must have a maximum distance where the POTENTIAL GRAVITATIONAL energy equals the kinetic energy of two lightest particles at near "c". and our "real" universe is smaller because I know more than 2 particles exist...\

Do you agree, or am I missing something?

 

[JesseM]

So, I say that if we assume the entire Universe is reduced to two "lightest" particles, the Universe MUST have a MAXIMUM size because two of the lightest particles must have a maximum distance where the POTENTIAL GRAVITATIONAL energy equals the kinetic energy of two lightest particles at near "c". and our "real" universe is smaller because I know more than 2 particles exist...

What to you mean the "kinetic energy of two lightest particles at near c"? How near, exactly? No matter how large a finite energy E you choose, that will always be equal to the kinetic energy of the particles at some velocity v less than c, so it will always be possible to imagine the particles moving at a velocity less than c but larger than v where their kinetic energy would be greater than E. So, it's impossible to find any finite upper limit on the amount of kinetic energy those particles could have with a velocity sufficiently close to c.

 

[yogi]

A similar circumstance arises If the universe is considered to have critical density, and you imagine it is a black hole with all the matter concentrated at the Hubble center, a particle ejected from the concentrated center at a velocity approacing c would have zero velocity as it exited the observable universe - but this doesn't provide any information about the size of the universe beyond the Hubble sphere - the same is true working the dynamics in reverse - you cannot make any meaningful predicitions about the size of universe based upon KE derived from distances beyond the Hubble sphere

 

[JesseM]

A similar circumstance arises If the universe is considered to have critical density, and you imagine it is a black hole with all the matter concentrated at the Hubble center, a particle ejected from the concentrated center at a velocity approacing c would have zero velocity as it exited the observable universe

How do you figure? Are you getting this from a source or is it your own argument?

 

[Q-reeus]

..1. Imagine 2 particles (lightest particle with any finite mass)

Why 'lightest particle with mass' (presumably a neutrino)? How does this effect the argument?

2. separated by the the entire distance of the universe

What kind of 'universe' exactly? I'm guessing you are thinking of a flat Minkowski metric upon which these two particles just sit. Is that true? I have no idea what a 'universe' of two particles governed by GTR would look like, but nothing like the one we inhabit I would say!

3. the particles are attracted (due to gravity) to each other and will require an infinite amount of energy to reach full "light speed"

Sure, but where would the energy come from? Since the total initial energy of this system (two lightest particles at 'infinite' separation) is just 2*mc² (m being the mass of each 'lightest particle'), then by the conservation of energy all that can change as distance decreases is an exchange between potential and KE, the total remaining constant. you are I assume aware that as a classical force, gravitational attraction drops off as the inverse square of separation distance.

The particles will accelerate (increase mass due to relativity) as they move toward another and speed up. This will require an infinite amount of energy.

As per 3 above, how so? I suspect you are picturing a constant force of gravitational attraction independent of separation, which is far from true.

So, I say that if we assume the entire Universe is reduced to two "lightest" particles, the Universe MUST have a MAXIMUM size because two of the lightest particles must have a maximum distance where the POTENTIAL GRAVITATIONAL energy equals the kinetic energy of two lightest particles at near "c". and our "real" universe is smaller because I know more than 2 particles exist...\
Do you agree, or am I missing something?

A lot I would say. Clarify points 1-3 and your strange (to me at least) reasoning above should change somewhat!

 

[Dale]

two of the lightest particles must have a maximum distance where the POTENTIAL GRAVITATIONAL energy equals the kinetic energy of two lightest particles at near "c".

Hi AJustice, welcome to PF! Unfortunately for your analysis, the kinetic energy of a particle at near c is unbounded.

Re: your argument with your wife. I can't believe that you would argue with your wife about this kind of stuff. IMO, you really need to learn to pick your battles! Make sure you score some points losing this one.

 

[AJustice]

Point 1 (particle must have mass so the kinetic energy formula isn't equal to zero

Point 2 "The universe is in this case a large 3d volume that has a limited diameter equal to the distance of these two "lightest" particles at their maximum seperation

Point 3 Yes, I understand that gravity drops off... But, it never fades to zero. So, maybe the first 4 lightyears of separation stores 3MJ of potential energy and the next 4 light years only stores 0.0003MJ. BUT, there is still energy stored by seperating two particles by a distance. ALSO, if I remember correctly, as the particles head towards one another, (approaching the speed of light), the gravitational force between the two increases. (their gravity, not mass, increases)

So, the point is, no matter how infinitely small, a near infinite (but finite) distance will store a tremendous amount of energy.

If I take two particles I CANNOT separate them by a distance that would equate to a potential energy greater than the maximum kinetic energy (both particles at the speed of light)

 

[DrGreg]

If I take two particles I CANNOT separate them by a distance that would equate to a potential energy greater than the maximum kinetic energy (both particles at the speed of light)

Is the source of your confusion that you think the Newtonian formula for kinetic energy ½mv² applies here? It doesn't. The relativistic formula for kinetic energy is

$$\left( \frac{1}{\sqrt{1-v^2/c^2}}-1\right) mc^2$$​

If you try to put v=c and m>0 in that, you get ∞, which is the mathematics telling you that you've tried to do something impossible. In particular, for a given fixed m, there is no maximum value to the expression; you can make it as large as you like.

 

[AJustice]

Is the source of your confusion that you think the Newtonian formula for kinetic energy ½mv² applies here? It doesn't. The relativistic formula for kinetic energy is

$$\left( \frac{1}{\sqrt{1-v^2/c^2}}-1\right) mc^2$$​

If you try to put v=c and m>0 in that, you get ∞, which is the mathematics telling you that you've tried to do something impossible. In particular, for a given fixed m, there is no maximum value to the expression; you can make it as large as you like.

DrGreg, I think I am arguing the opposite side of the same coin. I am saying since there it would take an infinite amount of energy to reach "C" (you can't ever do it) then if we go to .9999999999999999999999999c work the equation equal to the potential energy equation... you should have the size of the universe (since a lot of energy is tied up in the other REAL masses that we know exist the .99999...should be a reasonable estimate)

 

[JesseM]

DrGreg, I think I am arguing the opposite side of the same coin. I am saying since there it would take an infinite amount of energy to reach "C" (you can't ever do it) then if we go to .9999999999999999999999999c work the equation equal to the potential energy equation... you should have the size of the universe (since a lot of energy is tied up in the other REAL masses that we know exist the .99999...should be a reasonable estimate)

Why should the size of the universe be the distance where the gravitational potential is equal to the kinetic energy at .9999999999999999999999999c? This energy would be very small compared to the kinetic energy at .999999999999999999999999999999999999999999999999c for example, which itself would be very small compared to the kinetic energy at an even larger fraction of c, etc.

The point is, by picking larger and larger fractions of light speed you get larger and larger values of the kinetic energy of a given particle at that speed, with no upper limit. And thus there is no upper limit on the distance at which the gravitational potential is equal to kinetic energy at an arbitrarily large fraction of light speed. For any distance you can think of, if you calculate the potential energy between two particles at that distance, I can come up with a speed such that the kinetic energy of the particles at my chosen speed will be 10000 times larger.

 

[AJustice]

The idea isn't to come up with a value, but a thought system that shows that unless there is an infinite amount of energy, then the universe MUST be finite.

 

[JesseM]

The idea isn't to come up with a value, but a thought system that shows that unless there is an infinite amount of energy, then the universe MUST be finite.

In a universe with an infinite amount of matter (which is what is usually meant by an "infinite universe"), the energy would be infinite too, that just follows from E=mc^2.

 

[Dale]

AJustice, your argument is wrong. There is no maximum ke, therefore the size cannot be constrained by a nonexistent maximum ke.

 

[PAllen]

It seems to me a really basic misunderstanding of the original post is that two particles at 'infinity' will gain infinite energy by the time they eventually collide, infinitely later. Instead, assuming e.g. baseballs, they will have an extremely small amount of energy, based the mutual escape velocity of two baseballs. If elementary particles, you've got to bring quantum mechanics into the picture (a zero radius particle of nonzero mass is a classical contradiction). They would interact with either in various ways long before any possibility of infinite density; and if you assume non-zero radius, then, even without quantum mechanics you just have the baseball scenario.

Thus the whole construction is simply wrong from the start.

 

[yogi]

How do you figure? Are you getting this from a source or is it your own argument?

Use the relationship [GM/R(c^2)] = 1 within the limits of experimental error

 

[JesseM]

Use the relationship [GM/R(c^2)] = 1 within the limits of experimental error

That doesn't really help explain much of anything you said, you'll need to provide a more detailed explanation of the various statements you made in that previous post. First of all, is that equation supposed to be the relationship between R and M for the Schwarzschild radius? If so it's not right, it should be [GM/R(c^2)] = 1/2. But what does this have to do with the rest of your statements? Are you saying that the radius of the observable universe is similar to the Schwarzschild radius of the mass contained in it? Even if this were true it would not make the universe a black hole, see the first section here. And even if we were in a universe-sized black hole, it would not make sense to say "a particle ejected from the concentrated center at a velocity approacing c would have zero velocity as it exited the observable universe" since it's impossible for anything to move in the outward direction once inside a black hole (it could move outward if the universe were a white hole, a possibility also discussed in the FAQ link above, but even in that case I don't understand where you would get the idea that a particle ejected from the central singularity "at a velocity approaching c" would "have zero velocity" as it passed outward through the event horizon).