# Naked Science Forum

## Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: jeffreyH on 08/03/2015 18:37:14

Title: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 08/03/2015 18:37:14
For a mass m we can define a function ve = f(r) for escape velocity. We should then also be able to define a function td = f(r). Can we then define a function td = f(ve)? What would this relate to? Everything is relative so there is no universal frame of reference.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 09/03/2015 01:49:50
I found these two consecutive posts on physicsforums.

1) The time dilation caused by gravity on the surface of a planet is equal to the time dilation for an object moving at the planet's escape velocity in space. This can be proved using the Schwarzschild metric. GR doesn't explain why this is true. It seems to be an odd coincidence.

2) Meaning, I assume, in free space, far away from all gravitating bodies?

Is this correct?
Title: Re: Can we relate time dilation to escape velocity?
Post by: Ethos_ on 09/03/2015 21:03:59
I found these two consecutive posts on physicsforums.

1) The time dilation caused by gravity on the surface of a planet is equal to the time dilation for an object moving at the planet's escape velocity in space. This can be proved using the Schwarzschild metric. GR doesn't explain why this is true. It seems to be an odd coincidence.

2) Meaning, I assume, in free space, far away from all gravitating bodies?

Is this correct?
It's really very simple Jeff. Think of this in terms of the black hole. We are all familiar with the escape velocity of a black hole as being equal to c.

Understanding this, the fact that escape velocity for any mass in question is caused by the gravitational influence, and as such is directly proportional  to the time dilation associated with that mass. That fact should come as no surprise.

Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 09/03/2015 21:58:38
I found these two consecutive posts on physicsforums.

1) The time dilation caused by gravity on the surface of a planet is equal to the time dilation for an object moving at the planet's escape velocity in space. This can be proved using the Schwarzschild metric. GR doesn't explain why this is true. It seems to be an odd coincidence.

2) Meaning, I assume, in free space, far away from all gravitating bodies?

Is this correct?
It's really very simple Jeff. Think of this in terms of the black hole. We are all familiar with the escape velocity of a black hole as being equal to c.

Understanding this, the fact that escape velocity for any mass in question is caused by the gravitational influence, and as such is directly proportional  to the time dilation associated with that mass. That fact should come as no surprise.

Well, that being said, I am asking if the same escape velocity for any size of mass will give the same time dilation. This requires a variation in density.

EDIT: This should relate to black hole entropy.

http://en.wikipedia.org/wiki/Black_hole_thermodynamics

This also involves the Bekenstein Bound

http://en.wikipedia.org/wiki/Bekenstein_bound
Title: Re: Can we relate time dilation to escape velocity?
Post by: Ethos_ on 09/03/2015 22:47:02

This requires a variation in density.
That's very true Jeff but as densities are different for rocky planets like earth, and gas giants like Jupiter, these differences in density also effect where the surface of each is found. And consequently, the radius of each which is used to calculate the escape velocities of those bodies.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 09/03/2015 23:58:32
Distance traveled in free fall is:

s = (1/2)gt^2

Since g = GM/r^2

We can say that

S = (1/2)(GM/r^2)t^2

Then 2Sr/t^2 = GM/r

Ve = SQRT(2GM/r)

So:

4Sr/t^2 = 2GM/r

The factor of 4 is now present

SQRT(4Sr/t^2) = SQRT(2GM/r)

Since g = GM/r^2

Then

Ve = SQRT(2gr) for any mass.
Title: Re: Can we relate time dilation to escape velocity?
Post by: alancalverd on 10/03/2015 00:16:38
But that all assumes point masses.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 10/03/2015 00:25:55
But that all assumes point masses.

Yes and that is a major problem. Which highlights just why GR isn't a piece of cake.
Title: Re: Can we relate time dilation to escape velocity?
Post by: Bill S on 10/03/2015 00:53:33
Quote from: Ethos
We are all familiar with the escape velocity of a black hole as being equal to c.

If that were the case, wouldn't light escape?
Title: Re: Can we relate time dilation to escape velocity?
Post by: Ethos_ on 10/03/2015 01:05:43
Quote from: Ethos
We are all familiar with the escape velocity of a black hole as being equal to c.

If that were the case, wouldn't light escape?
Yes of course...........I should have said "escape velocity greater than c." But greater by only the smallest degree.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 10/03/2015 11:59:42
With Ve = SQRT(2gr) if we hold g constant and vary r as in a change in density then we have a differing value for Ve. So gravitational acceleration and escape velocity are not proportional. This means that while for the earth the value of g at the surface is much less than Ve there can be situations in which g is greater than Ve near very dense objects. In theory this can result in superluminal acceleration.
Title: Re: Can we relate time dilation to escape velocity?
Post by: Ethos_ on 10/03/2015 15:11:26
With Ve = SQRT(2gr) if we hold g constant and vary r as in a change in density then we have a differing value for Ve. So gravitational acceleration and escape velocity are not proportional. This means that while for the earth the value of g at the surface is much less than Ve there can be situations in which g is greater than Ve near very dense objects. In theory this can result in superluminal acceleration.
I think you've misunderstood my use of the word proportional Jeff. I was not saying they were equal, Webster's defines proportional as:

"to arrange the parts of (a whole) so as to be harmonious. a ratio"

Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 10/03/2015 17:44:43
With Ve = SQRT(2gr) if we hold g constant and vary r as in a change in density then we have a differing value for Ve. So gravitational acceleration and escape velocity are not proportional. This means that while for the earth the value of g at the surface is much less than Ve there can be situations in which g is greater than Ve near very dense objects. In theory this can result in superluminal acceleration.
I think you've misunderstood my use of the word proportional Jeff. I was not saying they were equal, Webster's defines proportional as:

"to arrange the parts of (a whole) so as to be harmonious. a ratio"

Well there is a proportionality and it is balanced but not like the inverse square nature of the gravitational field. Add time dilation to the mix and stir. Can we say time dilation is a function of Ve, g or both?
Title: Re: Can we relate time dilation to escape velocity?
Post by: JohnDuffield on 10/03/2015 17:54:08
It isn't a function of g, because that denotes the "local slope" of gravitational potential. Gravitational time dilation denotes the depth of gravitational potential. And to escape it, you need to "take a run at it", wherein your escape velocity is related to the gravitational time dilation.

Quote from: jeffreyH
The time dilation caused by gravity on the surface of a planet is equal to the time dilation for an object moving at the planet's escape velocity in space. This can be proved using the Schwarzschild metric. GR doesn't explain why this is true. It seems to be an odd coincidence.
It's no coincidence. The thing is, GR doesn't actually explain why matter falls down. It doesn't actually say why an object acquires some given speed, which is escape velocity when you flip things round. However it's very easy to understand if you think about the wave nature of matter and stuff electron diffraction and spin. Just simplify the electron to light going round and round, then simplify it further to light going round a square path. The horizontals bend in the gravitational gradient, and the electron falls down. It's similar if you accelerate the electron in gravity-free space. Sadly you never seem to see any texts which combine relativity with the wave nature of matter.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 10/03/2015 21:51:05
It isn't a function of g, because that denotes the "local slope" of gravitational potential. Gravitational time dilation denotes the depth of gravitational potential. And to escape it, you need to "take a run at it", wherein your escape velocity is related to the gravitational time dilation.

Quote from: jeffreyH
The time dilation caused by gravity on the surface of a planet is equal to the time dilation for an object moving at the planet's escape velocity in space. This can be proved using the Schwarzschild metric. GR doesn't explain why this is true. It seems to be an odd coincidence.
It's no coincidence. The thing is, GR doesn't actually explain why matter falls down. It doesn't actually say why an object acquires some given speed, which is escape velocity when you flip things round. However it's very easy to understand if you think about the wave nature of matter and stuff electron diffraction and spin. Just simplify the electron to light going round and round, then simplify it further to light going round a square path. The horizontals bend in the gravitational gradient, and the electron falls down. It's similar if you accelerate the electron in gravity-free space. Sadly you never seem to see any texts which combine relativity with the wave nature of matter.

Of course it has to be a function of g. If you are stationary on the earth you are experiencing time dilation. You are not accelerating away from it. The velocity involved in escaping the field is due to kinetic energy. If the gravitational field were absent the time dilation would be due to the velocity. At a stationary position on the earth there is still a potential for acceleration. However there is NO potential for escape unless there is an impetus away from the surface. This will INDUCE a time dilation. Without the impetus there is no dilation due to Ve. Can't you see that? You only have the value of g operating.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 10/03/2015 23:34:14
What are the consequences of reformulating the Ve equation this way?

c = SQRT(2gr)

This is then simplified to:

g = c^2/2r

EDIT: Of course this should be:

g = c^2/2rs

This is likely the reason for the survival of the G2 gas cloud.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 11/03/2015 00:58:28
In the proper Schwarzschild form we have:

a = (GM)/(r^2*SQRT[1-rs/r])

Where the above is:

g = c^2/2rs

In the Schwarzschild form the acceleration goes infinite at the event horizon.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 11/03/2015 01:53:06
Considering a point an infinitesimal distance from rs denoted by n we can then produce the following,

g = c^2/(2[rs+n]*SQRT(1-rs/[rs+n]))

The value of n can then be increased as needed to plot the gradient of g away from the event horizon.
Title: Re: Can we relate time dilation to escape velocity?
Post by: JohnDuffield on 11/03/2015 14:11:00
Of course it has to be a function of g.
No it isn't. See this depiction of gravitational potential:

CCASA image by AllenMcC, see Wikipedia (http://commons.wikimedia.org/wiki/File:GravityPotential.jpg).

The time dilation is represented by the depth of potential, how low in the plot you are, whilst g is the slope of the plot. Note that there's an inflection, so g at one elevation is the same as g at another. You can find two places on the plot where you can draw the same tangent.

If you are stationary on the earth you are experiencing time dilation.
Yes.

You are not accelerating away from it.
Yes.

The velocity involved in escaping the field is due to kinetic energy.
Yes, and in your ascending rocket going faster and faster you are swapping gravitational time dilation for special-relativistic time dilation.

If the gravitational field were absent the time dilation would be due to the velocity.
Yes.

At a stationary position on the earth there is still a potential for acceleration. However there is NO potential for escape unless there is an impetus away from the surface. This will INDUCE a time dilation.
Yes. And as you ascend away from the surface you reduce the gravitational time dilation.

Without the impetus there is no dilation due to Ve. Can't you see that?
Yes.

You only have the value of g operating.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 11/03/2015 14:56:45
Of course it has to be a function of g.
No it isn't. See this depiction of gravitational potential:

CCASA image by AllenMcC, see Wikipedia (http://commons.wikimedia.org/wiki/File:GravityPotential.jpg).

The time dilation is represented by the depth of potential, how low in the plot you are, whilst g is the slope of the plot. Note that there's an inflection, so g at one elevation is the same as g at another. You can find two places on the plot where you can draw the same tangent.

If you are stationary on the earth you are experiencing time dilation.
Yes.

You are not accelerating away from it.
Yes.

The velocity involved in escaping the field is due to kinetic energy.
Yes, and in your ascending rocket going faster and faster you are swapping gravitational time dilation for special-relativistic time dilation.

If the gravitational field were absent the time dilation would be due to the velocity.
Yes.

At a stationary position on the earth there is still a potential for acceleration. However there is NO potential for escape unless there is an impetus away from the surface. This will INDUCE a time dilation.
Yes. And as you ascend away from the surface you reduce the gravitational time dilation.

Without the impetus there is no dilation due to Ve. Can't you see that?
Yes.

You only have the value of g operating.

You have just contradicted yourself and described time dilation as a function of g.
Title: Re: Can we relate time dilation to escape velocity?
Post by: David Cooper on 11/03/2015 18:41:23
It's no coincidence. The thing is, GR doesn't actually explain why matter falls down. It doesn't actually say why an object acquires some given speed, which is escape velocity when you flip things round. However it's very easy to understand if you think about the wave nature of matter and stuff electron diffraction and spin. Just simplify the electron to light going round and round, then simplify it further to light going round a square path. The horizontals bend in the gravitational gradient, and the electron falls down. It's similar if you accelerate the electron in gravity-free space. Sadly you never seem to see any texts which combine relativity with the wave nature of matter.

That's a useful way of looking at things. Where do you get your knowledge from, because I'd like to explore the source.
Title: Re: Can we relate time dilation to escape velocity?
Post by: JohnDuffield on 12/03/2015 17:18:43
That's a useful way of looking at things. Where do you get your knowledge from, because I'd like to explore the source.
A whole rack of different places, ranging from old papers by the likes of Maxwell and Einstein, to newer papers and articles  that tend not to get much publicity. See for example http://arxiv.org/abs/physics/0512265 and S A Goudsmit talking about the discovery of electron spin (http://lorentz.leidenuniv.nl/history/spin/goudsmit.html): "When the day came I had to tell Uhlenbeck about the Pauli principle - of course using my own quantum numbers - then he said to me: "But don't you see what this implies? It means that there is a fourth degree of freedom for the electron. It means that the electron has a spin, that it rotates". Also see magnetic moment (http://en.wikipedia.org/wiki/Electron_magnetic_moment#Magnetic_moment_of_an_electron) and the Einstein-de Haas effect (http://en.wikipedia.org/wiki/Einstein%E2%80%93de_Haas_effect) which "demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics". However if you were to ask a contemporary particle physicist about this, he might say the electron is a point particle, that its spin is not a real rotation, and that gravity is all down to gravitons flitting back and forth.

NB: note that only the horizontals bend downwards, which is why  light is deflected twice as much as matter (http://www.astro.ucla.edu/~wright/deflection-delay.html).
Title: Re: Can we relate time dilation to escape velocity?
Post by: David Cooper on 12/03/2015 19:43:58
Thanks John - I'll follow all of those up.
Title: Re: Can we relate time dilation to escape velocity?
Post by: JohnDuffield on 12/03/2015 22:21:21
There's loads more interesting material out there. IMHO an interesting read is On Vortex Particles (http://www.scribd.com/doc/68152826/On-Vortex-Particles-Fiasco-Press-Journal-of-Swarm-Scholarship#scribd) by David St John. He's a recent physics PhD. It isn't peer reviewed or anything, but it gives a wealth of information going back to Thomson and Tait, which arguably goes back to Maxwell's theory of molecular vortices (http://en.wikisource.org/wiki/On_Physical_Lines_of_Force). The gist of it is that the electron is a spinor (http://en.wikipedia.org/wiki/Spinor) which is akin to a cyclone. The positron is a similar spinor with the opposite chirality which is akin to an anticyclone, and counter-rotating vortices attract. I really ought to write a paper on it. I was talking to the SPIE (http://spie.org/app/search/browse/?Terms=duffield)  guys about this, but I can't go to San Diego in August.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 12/03/2015 22:37:25
On the Hubius Helix.

Read, enjoy and maybe provide all the proofs required.

http://www.quora.com/What-do-physicists-think-of-the-hubius-helix-model-of-electron-structure
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 12/03/2015 23:00:04
Strangely John there is a Mr John Duffield and a Mr Stuart J Duffield at SPIE. Which one are you? Doesn't this get confusing over who wrote what. One may easily think one Duffiled wrote something when in fact it was the other. That would be unfortunate.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 12/03/2015 23:18:45
John you aren't going to claim to be an authority on spinors are you? How much group theory do you know? This is an honest question and I think it deserves an honest answer. If you are going to be making these claims and representing yourself as an authority, where people might actually be convinced that they are learning established physics from you, then put up or shut up. It is hard enough to learn physics without being waylaid by bogus claims. You do every enthusiastic amateur an injustice by doling out spurious information simply to boost your status in the eyes of others.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 12/03/2015 23:45:21
Last night I watch the Horizon programme on the BBC about the search for the gravitational waves from the big bang. Alan Guth was one of the physicists included. He propose the theory of inflation right at the start of his career. Finding those waves will tell us a lot more about the big bang and validate Alan's theory. The guys set up telescopes at the pole and spent 3 years recording CMBR from a dark spot in the sky. Then after 3 years they found the data wasn't accurate enough. They then went away and built a better telescope and spent several more years looking for the b mode signal of the early gravitational waves. They found something that looked like the signal and spent more time analyzing it and ruling things out until they were confident in announcing it. Then it was shown that at least 50% of it and maybe 100% of the signal was due to dust. That is physics.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 12/03/2015 23:48:51
For anyone interested in the signal there is information here;

http://background.uchicago.edu/~whu/intermediate/Polarization/polar6.html
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 13/03/2015 00:16:00
The thing that crosses my mind about the equation

g = c^2/(2[rs+n]*SQRT(1-rs/[rs+n]))

is how would this relate to the Bekenstein bound

http://en.wikipedia.org/wiki/Bekenstein_bound

and the chandrasekhar limit

http://en.wikipedia.org/wiki/Chandrasekhar_limit

If a connection to the black hole entropy can be found that would be a positive step. With the chandrasekhar limit it may be possible to put a lower and upper bound on possible black hole masses. In respect of the big bang this could be very important if the idea of a singularity is to hold.
Title: Re: Can we relate time dilation to escape velocity?
Post by: JohnDuffield on 13/03/2015 01:01:02
Jeffrey: David Simmons-Duffin was being overly hostile. Take a look at the picture on the quora page. Then note that the Dirac spinor (http://en.wikipedia.org/wiki/Dirac_spinor) is a bispinor, and have a look at Dirac's belt (http://www.mathpages.com/home/kmath619/kmath619.htm) on Mathspages:

"In contrast, the Mobius strip is a non-orientable surface, because a right-handed figure, moved continuously around the loop until arrive back at its starting point, becomes left-handed. An object must be translated around the loop twice in order to be restored to its original position and chirality. In this sense a Mobius strip is reminiscent of spin-1/2 particles in quantum mechanics, since such particles must be rotated through two complete rotations in order to be restored to their original state...

I'm John Duffield. I don't know any Stuart J Duffield. I'm not claiming to be an authority on spinors or group theory, I'm just telling you about things that aren't in the popscience/student books/articles/etc you've been reading. Do not think that something is "spurious information" or isn't "established physics" just because you don't know about it. The vast bulk of what I tell you about is established physics. I didn't invent the word spinor, and I didn't write the Wikipedia article or put this picture on it:

Quote
Last night I watch the Horizon programme on the BBC about the search for the gravitational waves from the big bang. Alan Guth was one of the physicists included. He propose the theory of inflation right at the start of his career. Finding those waves will tell us a lot more about the big bang and validate Alan's theory. The guys set up telescopes at the pole and spent 3 years recording CMBR from a dark spot in the sky. Then after 3 years they found the data wasn't accurate enough. They then went away and built a better telescope and spent several more years looking for the b mode signal of the early gravitational waves. They found something that looked like the signal and spent more time analyzing it and ruling things out until they were confident in announcing it. Then it was shown that at least 50% of it and maybe 100% of the signal was due to dust. That is physics.
The Horizon producers pulled their punches, and didn't say anything about the orchestrated hype that put people's back up. See Physicist Paul Steinhardt Slams Inflation, Cosmic Theory He Helped Conceive (http://blogs.scientificamerican.com/cross-check/2014/12/01/physicist-paul-steinhardt-slams-inflation-cosmic-theory-he-helped-conceive/). And note this: that "first light" dates from 300,000 years after the big bang. How can that tell you anything about the first nanosecond after the big bang? The universe was a seething maelstrom of plasma for a third of a million years. It would be like putting those sonic standing waves into a blender.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 13/03/2015 01:18:29
John, how do you validate anything if you don't understand it? You now say you are not an authority on spinors. What exactly DO you know about them?

Bispinor on wikipedia.

http://en.wikipedia.org/wiki/Bispinor

Have you even looked at the equations in there? You have group mapping, transformations and the SO(3;1) group all mentioned.

You can find information on orthogonal groups here:

http://en.wikipedia.org/wiki/Orthogonal_group

"Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication, and an orthogonal matrix is a real matrix whose inverse equals its transpose."

"The determinant of an orthogonal matrix being either 1 or −1, an important subgroup of O(n) is the special orthogonal group, denoted SO(n), of the orthogonal matrices of determinant 1. This group is also called the rotation group, because, in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see SO(2), SO(3) and SO(4)."

What about symmetries? Does that ring any bells?
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 13/03/2015 01:22:30
Now then John what type of matrix has its inverse equaling its transpose?
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 13/03/2015 20:13:04
Jeffrey: David Simmons-Duffin was being overly hostile. Take a look at the picture on the quora page. Then note that the Dirac spinor (http://en.wikipedia.org/wiki/Dirac_spinor) is a bispinor, and have a look at Dirac's belt (http://www.mathpages.com/home/kmath619/kmath619.htm) on Mathspages:

"In contrast, the Mobius strip is a non-orientable surface, because a right-handed figure, moved continuously around the loop until arrive back at its starting point, becomes left-handed. An object must be translated around the loop twice in order to be restored to its original position and chirality. In this sense a Mobius strip is reminiscent of spin-1/2 particles in quantum mechanics, since such particles must be rotated through two complete rotations in order to be restored to their original state...

I'm John Duffield. I don't know any Stuart J Duffield. I'm not claiming to be an authority on spinors or group theory, I'm just telling you about things that aren't in the popscience/student books/articles/etc you've been reading. Do not think that something is "spurious information" or isn't "established physics" just because you don't know about it. The vast bulk of what I tell you about is established physics. I didn't invent the word spinor, and I didn't write the Wikipedia article or put this picture on it:

Quote
Last night I watch the Horizon programme on the BBC about the search for the gravitational waves from the big bang. Alan Guth was one of the physicists included. He propose the theory of inflation right at the start of his career. Finding those waves will tell us a lot more about the big bang and validate Alan's theory. The guys set up telescopes at the pole and spent 3 years recording CMBR from a dark spot in the sky. Then after 3 years they found the data wasn't accurate enough. They then went away and built a better telescope and spent several more years looking for the b mode signal of the early gravitational waves. They found something that looked like the signal and spent more time analyzing it and ruling things out until they were confident in announcing it. Then it was shown that at least 50% of it and maybe 100% of the signal was due to dust. That is physics.
The Horizon producers pulled their punches, and didn't say anything about the orchestrated hype that put people's back up. See Physicist Paul Steinhardt Slams Inflation, Cosmic Theory He Helped Conceive (http://blogs.scientificamerican.com/cross-check/2014/12/01/physicist-paul-steinhardt-slams-inflation-cosmic-theory-he-helped-conceive/). And note this: that "first light" dates from 300,000 years after the big bang. How can that tell you anything about the first nanosecond after the big bang? The universe was a seething maelstrom of plasma for a third of a million years. It would be like putting those sonic standing waves into a blender.

That article correctly states that Paul Steinhardt helped refine the theory years after Alan Guth conceived it. Don't put your own slant on things and try to rewrite history. So what if he objects to it. HE has a right to. Someone may well come up with a better theory. That is what physics is all about.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 13/03/2015 20:46:40
If we take the mass of the chandrasekhar limit and then calculate its Schwarzschild radius an equation can be formed thus:

g = c^2/[2rs+L/2]

The value for g that we then get is 299788351 which just about equals the speed of light. To find the lower limit of a feasible black hole we need to vary the parameters until this acceleration is as near to the surface of the event horizon as it is possible to get. This may or may not equal 1 Planck length.

CORRECTION: c/2 has been modified to L/2 to represent the distance traveled by light in one 1/2 second. You just can't add a velocity to a distance.
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 15/03/2015 20:01:26
If the limit on the lowest mass of a black hole coincides with the acceleration at one Planck length from the horizon being equal to the speed of light then that would relate directly to black hole entropy and the Bekenstein bound.
Title: Re: Can we relate time dilation to escape velocity?
Post by: SorryDnoodle on 15/03/2015 20:36:32
Quote from: Ethos
We are all familiar with the escape velocity of a black hole as being equal to c.

If that were the case, wouldn't light escape?
Yes of course...........I should have said "escape velocity greater than c." But greater by only the smallest degree.

Would't escape velocity be proportional to the mass of the black hole, in other words; A very small black hole being very close to c but a super massive black hole very much great than c?

If not, why?
Title: Re: Can we relate time dilation to escape velocity?
Post by: jeffreyH on 15/03/2015 21:04:31
Quote from: Ethos
We are all familiar with the escape velocity of a black hole as being equal to c.

If that were the case, wouldn't light escape?
Yes of course...........I should have said "escape velocity greater than c." But greater by only the smallest degree.

Would't escape velocity be proportional to the mass of the black hole, in other words; A very small black hole being very close to c but a super massive black hole very much great than c?

If not, why?

A black hole traps light at its event horizon. That is why it is called an event horizon because we cannot retrieve any information beyond the horizon. As the radius of the event horizon increases so does the mass but the density of the gravitational field decreases proportionally. If the escape velocity at the event horizon exceeded c then it would no longer be the event horizon. In that case the event horizon would be larger in radius.

You can find details here.