Asymptotic Freedom in QCD and...

[RajeshTrivedi]

This is trivially proven wrong. At r = 0, create a super-heavy mass. Your condition at r = 0 evaluates to infinity, meaning this super-heavy mass doesn't meet your condition of d(r) > 5.3 * 10^25/r^2. But, I can make it arbitrarily heavy, as long as I don't make it (unphysically) infinitely heavy. I can certainly make it heavy enough to generate an event horizon at small r, while still having an event horizon at the outer surface too. Conclusion: your condition is not enough to enforce that there only is an event horizon at the outer surface of the object.

So, do you agree that as long as density of any inner shell (when the object is just at EH) of radius r is < 5.3*10^25/r^2, all the inner points shall be out of their respective EHs?

 

[NotEinstein]

So, do you agree that as long as density of any inner shell (when the object is just at EH) of radius r is < 5.3*10^25/r^2, all the inner points shall be out of their respective EHs?

The task is to find an object that has an event horizon internally, but we have to avoid hitting this condition for any radius: d(r) > 5.3*10^25/r^2

Imagine an object with a radius larger than 1 meter. We're going to focus on the core up to 1 meter, and we're going to check whether there's an event horizon there. In other words, we are going to check whether $$r_s$$ is larger than 1 meter at an $$r$$ of 1 meter.

Let's calculate the mass of this core, by assuming it's right on the edge; in other words, all throughout the core the condition is met exactly. Now, writing out the maths is boring and silly-error prone, so we're going to be lazy and use one of Wolfram Alpha's widgets, the Spherical Integral Calculator:
http://www.wolframalpha.com/widgets/view.jsp?id=89c969c21b169fa996f899d9b2a98588

Let's put in the density condition: (5.3*10^25)/(rho^2) (in mathematics, they often use rho instead of r)
And integrate rho from 0.000001 to 1, phi from 0 to 2pi, and theta from 0 to pi (just look at: https://en.wikipedia.org/wiki/Spher..._and_differentiation_in_spherical_coordinates under "Thus, for example, a function...").

Result: 1.05*10^33 (kg)

This is the mass of the core; in other words, this is the maximum possible mass of an object of 1 meter that can still meet the condition.

To check whether this object has an event horizon, we calculate its Schwarzschild radius:
$$r_s=\frac{2GM}{c^2}$$

This gives:
$$r_s=1.485*10^-27\times1.05*10^33=1.6*10^6$$ (in meters)

Since $$1.6*10^6>1$$, we have $$r_s>r$$, and thus there's an event horizon around this object.

Note that we're missing the mass up to 0.000001 meter, because there's an infinity hiding there, so there's even more room for adding mass to this object.

Conclusion: something must have gone wrong in the derivation of the density condition, because there are objects possible that meet the condition while violating what it should stand for.

In other words: obviously I do not agree, no.

 

[RajeshTrivedi]

Result: 1.05*10^33 (kg)

Appears incorrect.
please apply proper conditions, from r = 0 to r = 1 meter and recalculate the mass of such sphere. Please give steps how you arrived at this mass from the given density profile. Because if I apply the formula used by me then the mass of a one meter sphere with such density profile comes at = 6.7 * 10^26 kg, and Rs of such object is also 1 meter; exactly tallying.

To check whether this object has an event horizon, we calculate its Schwarzschild radius:
$$r_s=\frac{2GM}{c^2}$$

This gives:
$$r_s=1.485*10^-27\times1.05*10^33=1.6*10^6$$ (in meters)

Since $$1.6*10^6>1$$, we have $$r_s>r$$, and thus there's an event horizon around this object.

Note that we're missing the mass up to 0.000001 meter, because there's an infinity hiding there, so there's even more room for adding mass to this object.

Conclusion: something must have gone wrong in the derivation of the density condition, because there are objects possible that meet the condition while violating what it should stand for.

In other words: obviously I do not agree, no.

Does not satisfy the condition.
Now assuming that your calculation is ok then also the object's Rs = 1600 km while object is of 1 meter size, that means it is not "just at EH", so you are failing the "just at EH" condition.

In other words: obviously I do not agree, no.

You will, once you understand it.

 

[RajeshTrivedi]

The point of argument is drifting, which is not so relevant here, the qualitative statement is as below:

1. In general when a celestial object is just of its Schwarzschild radius, then only the outer surface is just at its EH, all other inner layer points will be out of their respective (inner shell) schwarzschild radius.

2. As the object falls deeply beneath its EH, more and more layers fall inside their respective shell EH.

3. This is simply because the mass relationship with schwarzschild radius and actual radius varies with r and r^3 respectively.

Point#1 permits us to conclude that a photon emitted somewhere inside such core (when it is just at EH), can travel in any direction, it is not a compulsion that it will travel only towards the center. So somehow if we can generate photons at or around center of the core due to some mechanism, then these photons can travel away from the center towards outer surface (for how long and how much distance, not yet ascertained, that is the next step). However as long as the object is inside its EH, this photon cannot escape to the universe.

 

[NotEinstein]

Appears incorrect.
please apply proper conditions, from r = 0 to r = 1 meter and recalculate the mass of such sphere. Please give steps how you arrived at this mass from the given density profile. Because if I apply the formula used by me then the mass of a one meter sphere with such density profile comes at = 6.7 * 10^26 kg, and Rs of such object is also 1 meter; exactly tallying.

I put it in the calculator I linked, and this is the answer I got. If you claim it's incorrect, please explain to me what kind of mistake the calculator made.

Does not satisfy the condition.
Now assuming that your calculation is ok then also the object's Rs = 1600 km while object is of 1 meter size, that means it is not "just at EH", so you are failing the "just at EH" condition.

Exactly my point. It satisfies your condition on the density, but it doesn't satisfy your claim that such a density will be "just at EH", thus proving your derivation of that density to be faulty.

You will, once you understand it.

I can only, once you point out the supposed problem with the calculator.

The point of argument is drifting, which is not so relevant here, the qualitative statement is as below:

1. In general when a celestial object is just of its Schwarzschild radius, then only the outer surface is just at its EH, all other inner layer points will be out of their respective (inner shell) schwarzschild radius.

Except where the object generates two event horizons; one at its core, and one on its outer surface. This is of course easy to see: take an object that's fully underneath its event horizon, and add to it a spherical shell (radius outside the core's event horizon) that is heavy enough to generate its own event horizon.
(And before you complain; I'm still waiting for the (mathematical) description of a "realistic" density distribution you claim that needs to be used.)

2. As the object falls deeply beneath its EH, more and more layers fall inside their respective shell EH.

Agreed.​

3. This is simply because the mass relationship with schwarzschild radius and actual radius varies with r and r^3 respectively.

Not sure how that follows?​

Point#1 permits us to conclude that a photon emitted somewhere inside such core (when it is just at EH), can travel in any direction, it is not a compulsion that it will travel only towards the center. So somehow if we can generate photons at or around center of the core due to some mechanism, then these photons can travel away from the center towards outer surface (for how long and how much distance, not yet ascertained, that is the next step).

Probably true.

However as long as the object is inside its EH, this photon cannot escape to the universe.

​

Very true.

 

[RajeshTrivedi]

Except where the object generates two event horizons; one at its core, and one on its outer surface. This is of course easy to see: take an object that's fully underneath its event horizon, and add to it a spherical shell (radius outside the core's event horizon) that is heavy enough to generate its own event horizon.

Please show that for such object the density of any shell of size r will not exceed 5.3*10^25/r^2.

 

[RajeshTrivedi]

Point#1 permits us to conclude that a photon emitted somewhere inside such core (when it is just at EH), can travel in any direction, it is not a compulsion that it will travel only towards the center. So somehow if we can generate photons at or around center of the core due to some mechanism, then these photons can travel away from the center towards outer surface (for how long and how much distance, not yet ascertained, that is the next step).

Probably true.
​

So, you agree that it is probable, fine, thanks.
Now give an example when it is "false" without exceeding the density beyond 5.3*10^25/r^2 kg/m^3.

 

[NotEinstein]

Please show that for such object the density of any shell of size r will not exceed 5.3*10^25/r^2.

Irrelevant, as you said "In general", which means your condition need not apply.

So, you agree that it is probable, fine, thanks.

Under the assumption that you have an object where only the outer regions are within the event horizon, yes. I don't agree with the argumentation and derivation of point #1 (see above), but if you have such a situation, then photons in the inner parts of the object can probably travel somewhat outwards.

Now give an example when it is "false" without exceeding the density beyond 5.3*10^25/r^2 kg/m^3.

Again irrelevant, as you said "in general" in your point #1.

 

[RajeshTrivedi]

Irrelevant, as you said "In general", which means your condition need not apply.

Again irrelevant, as you said "in general" in your point #1.

So you cannot give an example? You object for the sake of objecting and when confronted, instead of backing out, you weave stories around your objections. Quite a dishonest chap you are!

My stand from the word go is very clear that as long as the density is less than 5.3*10^25/r^2 for inner shells of r (when an object is just at EH), all the inner points will be out of their respective EHs. You have not been able to counter it.

 

[NotEinstein]

So you cannot give an example?

I have given an example in post #205.

You object for the sake of objecting and when confronted, instead of backing out, you weave stories around your objections.

It’s you who’s dismissing my objections without addressing them properly. It’s you that dodging my objections. It’s you how apparently cannot back down when proven wrong.

Quite a dishonest chap you are!

If you believe that, feel free to report me to the moderation staff.

My stand from the word go is very clear that as long as the density is less than 5.3*10^25/r^2 for inner shells of r (when an object is just at EH), all the inner points will be out of their respective EHs. You have not been able to counter it.

I have given an example in post #205; an example that you have not been able to counter so far.

 

[NotEinstein]

It appears that RajeshTrivedi has abandoned his/her attempts to defend his/her position?

 

[Xelasnave.1947]

I do hope Rajesh is ok it is not like him to abandon something.
Alex

 

[NotEinstein]

I do hope Rajesh is ok it is not like him to abandon something.
Alex

Indeed, but perhaps I'm simply impatient. It is the time of year for close friends and family, and I doubt the Sciforums crowd can compete with those.

 

[RajeshTrivedi]

I do hope Rajesh is ok it is not like him to abandon something.
Alex

Actually I lost interest in this poster NotEinstein word manipulations.
He started playing with the words, almost trollish behaviour, offering no worthwhile argument.

Stay happy, healthy and prosper in new year 2018!!

 

[NotEinstein]

Actually I lost interest in this poster NotEinstein word manipulations.
He started playing with the words, almost trollish behaviour, offering no worthwhile argument.

As I thought: RajeshTrivedi is unable or unwilling to substantiate his/her claims, and unable or unwilling to disprove my objections. I guess everybody can draw their own conclusions.

 

[Xelasnave.1947]

Stay happy, healthy and prosper in new year 2018!!

Thank you and may I wish the same for you.
Keep going if you have time all interaction should be seen as beneficial.
Alex

 

[Xelasnave.1947]

I guess everybody can draw their own conclusions.

I think perhaps your earlier observation re the time of year will prove correct.
And may I also wish you a happy and prosperous 2018.
Alex

 

[NotEinstein]

I think perhaps your earlier observation re the time of year will prove correct.

I have to admit, it wasn't my observation originally (in another thread), so I deserve no credit there. But seeing the accusatory reply RajeshTrivedi just made, I highly doubt (s)he will continue conversering with me. Well, his/her loss!

And may I also wish you a happy and prosperous 2018.
Alex

And the same to you!