<p>I'm not sure what you mean by gravitational length contraction, and the time dilation formula you are using is only approximately correct for weak gravitational fields.</p>
<p>Observed neutron stars are indeed only a small factor larger than their Schwarzschild radii. If they were just a little smaller (say <span class="math-container">$&lt;1.4 r_s$</span>) then they would become unstable and likely collapse to become black holes.</p>
<p>Is this a coincidence? Well yes and no. It is the reason why we do not see neutron stars with masses of 3 times the sun or more. Such objects have indeed collapsed to have become black holes.</p>
<p>I suppose you could argue that it is an interesting twist of astrophysics that the massive stars that end up producing neutron stars finish their lives with iron cores that are too big to be supported by electron degeneracy pressure, but still small enough that they can avoid becoming a black hole. i.e. That the Chandrasekhar mass is smaller than the maximum mass for a neutron star. It wouldn't take much of a weakening of the strong nuclear force between closely packed neutrons to make forming neutron stars practically impossible. This is because neutron degeneracy pressure alone can only support neutron stars up to 0.75 solar masses and these could not be produced during stellar evolution without becoming stuck as stable white dwarfs. Anything more massive would then collapse directly to a black hole.</p>

CC BY-SA 4.0
2022-05-04T16:41:42.303Z
Of course there is gravitational length contraction. That's why Einstein talked about space-time warping and not just time warping. It's also why a photon doesn't lose or gain the ability to do work when it passes through a gravitational field. I would need to read up on electron and neutron degeneracy forces to understand your answer better, but I certainly find it interesting.
Alan Gee
CC BY-SA 4.0
2022-05-08T19:13:18.797Z
High mass stars are supported by nuclear fusion tho, which neutron stars are not, so I'm not sure what the twist is.
Daddy Kropotkin
CC BY-SA 4.0
2022-05-08T20:09:53.240Z
@DaddyKropotkin the cores of high-mass stars at the ends of their lives are supported (briefly) by electron degeneracy pressure. It is an accident of nuclear physics that the Chandrasekhar mass is smaller than the maximum mass of a neutron star.
ProfRob
CC BY-SA 4.0
2022-05-09T07:21:23.020Z
Oh I see, it was not clear to me that you meant at the end of nuclear fusion...
Daddy Kropotkin
CC BY-SA 4.0
2022-05-03T22:45:48.277Z
2022-05-08T20:07:42.870Z
2022-05-08T20:07:42.870Z
ProfRob
<p>Here's why I'm asking.</p>
<p>The gravitational time dilation factor at the surface of a star or planet can be expressed as <span class="math-container">$$gr/c^2$$</span></p>
<p>In theory, the same equation should apply to the gravitational length contraction factor.</p>
<p>If the gravitational length contraction factor reaches 1.0 then all lengths at the surface are reduced to 0 and the mass becomes invisible.</p>
<p>This will occur when <span class="math-container">$$gr = c^2$$</span></p>
<p>Now I have read that the surface gravity, g, on the largest known neutron star is about <span class="math-container">$$7*10^{12} m/s^2$$</span></p>
<p>Which means it should become invisible when <span class="math-container">$$r=c^2/7*10^{12}$$</span></p>
<p>Or <span class="math-container">$$r=12.84km$$</span></p>
<p>But that's the approximate radius of the largest known neutron star. So, in theory, it is on the verge of invisibility.</p>
<p>If it was any bigger it would probably be seen as a black hole.</p>
<p><strong>UPDATE</strong></p>
<p>As my formula for Gravitational Length Contraction has been questioned, I'd like to explain it at least, without attempting to prove it:</p>
<p>In that formula, <span class="math-container">$g$</span> is the rate of length contraction from the centre to the surface of the mass in units of <span class="math-container">$m/s^2$</span>.</p>
<p>But the contraction happens at the speed of light, so the <span class="math-container">$c^2$</span> term is a conversion from time squared to length squared.</p>
<p>So <span class="math-container">$g/c^2$</span> is the rate of length contraction from the centre to the surface in units of <span class="math-container">$m/m^2$</span> or <span class="math-container">$m/m$</span> per <span class="math-container">$m$</span> radius.</p>
<p>Multiplying this by the radius gives the length contraction factor at the surface in <span class="math-container">$m/m$</span>.</p>
<p><strong>UPDATE END</strong></p>
<p>So is a black hole a neutron star with a gravitational length contraction factor of 1.0, or is it pure coincidence that the largest known neutron star has a gravitational length contraction factor very close to 1.0?</p>

CC BY-SA 4.0
2022-05-03T17:36:57.443Z
Are you sure about your formula? How did you derive it?
Anders Sandberg
CC BY-SA 4.0
2022-05-03T22:37:57.883Z
Your time dilation formula (for $1-t_0/t_f$) is an approximation. It's ok when $gr/c^2$ is small, but not when GR effects become significant. Instead, use $t_0 = t_f\sqrt{1 - \frac{r_s}{r}}$, where $r_s=2GM/c^2$, which can be found at https://en.wikipedia.org/wiki/Gravitational_time_dilation#Outside_a_non-rotating_sphere
PM 2Ring
CC BY-SA 4.0
2022-05-03T23:06:56.923Z
Here's a Schwarzschild radius & time dilation [calculator](https://sagecell.sagemath.org/?z=eJxVjsFqwzAQRO_6ioFAbbVJI9kOjgyGntpTT7k7CEshAstOJZnifn3XKZRWoMPO7NuZDU799VOHr9hf3WAQtHFzxAOS8xbGDTq5aWQbvL1juuA0j3AjfFfuY1cwP5_QQpZFLYuqEqKSz6pU5RGPkKJTTM-rXamDqo-1qIVgPQmFUrUqqsORsRc3Jht0n5ixF5xz3-ZZnAYd4HWMNmZbSL5FINm4mPTYW-Sek1wKIcjRc5rO883oZNtXPUTLGwZ6Ia5B1MOvn2ru0VPh1boFCs2zcI4N3Qlxi8xn_G4tBBG5R7iPydAcP0LKJXZY-F88mQYgPpkfeV1dt5L5HfMFTzBdwengv2i5I5pYw78B52Bjtw==&lang=sage). You can put expressions in the input fields,  eg `10*au`  gives a distance of 10  AU.
PM 2Ring
CC BY-SA 4.0
2022-05-05T07:59:41.350Z
@PM2Ring Schwartzchild got this wrong. Mass is decelerated away from the surface by gravity and therefore has an escape velocity. Light is accelerated away from the surface by gravity, and can therefore escape regardless of the surface gravity. This not so subtle difference between mass and light appears to have been quite readily ignored by Schwartzchild and many others since.
Alan Gee
CC BY-SA 4.0
2022-05-07T23:19:16.470Z
@AlanGee - Sorry, but this does not make much sense. Light is not "accelerated away from the surface": unless pointed radially it follows a curved null geodesic and as the Schwarzschild radius is approached most such geodesics end up captured by the surface. And "rate of length contraction factor" is an entirely nonstandard term that doesn't make much sense. Are you trying to measure the difference in radial geodesic distance from coordinate distance?
Anders Sandberg
CC BY-SA 4.0
2022-05-10T12:18:37.913Z
My understanding of gravity is based on the equation of free fall:- $dE/ds = -dp/dt$ that is applicable to both mass and light. For light, this means that the downward force generated by the energy gradient due to time dilation must be opposed by an equal and opposit upward force generated by a momentum gradient due to length contraction. Given that, it is easy to see that light is decelerated towards the surface, since its wavelength is reducing and its period is increasing in that direction. Of course if you measure it on any geodesic it will always look the same.
Alan Gee
<black-hole><gravity><neutron-star>
Is a black hole really a neutron star with a gravitational time dilation factor of 1.0
Alan Gee
# Is a black hole really a neutron star with a gravitational time dilation factor of 1.0

(Asked by: username_0 on 2022-05-03)

Here's why I'm asking.

The gravitational time dilation factor at the surface of a star or planet can be expressed as $$gr/c^2$$

In theory, the same equation should apply to the gravitational length contraction factor.

If the gravitational length contraction factor reaches 1.0 then all lengths at the surface are reduced to 0 and the mass becomes invisible.

This will occur when $$gr = c^2$$

Now I have read that the surface gravity, g, on the largest known neutron star is about $$7\*10^{12} m/s^2$$

Which means it should become invisible when $$r=c^2/7\*10^{12}$$

Or $$r=12.84km$$

But that's the approximate radius of the largest known neutron star. So, in theory, it is on the verge of invisibility.

If it was any bigger it would probably be seen as a black hole.

**UPDATE**

As my formula for Gravitational Length Contraction has been questioned, I'd like to explain it at least, without attempting to prove it:

In that formula, $g$ is the rate of length contraction from the centre to the surface of the mass in units of $m/s^2$.

But the contraction happens at the speed of light, so the $c^2$ term is a conversion from time squared to length squared.

So $g/c^2$ is the rate of length contraction from the centre to the surface in units of $m/m^2$ or $m/m$ per $m$ radius.

Multiplying this by the radius gives the length contraction factor at the surface in $m/m$.

**UPDATE END**

So is a black hole a neutron star with a gravitational length contraction factor of 1.0, or is it pure coincidence that the largest known neutron star has a gravitational length contraction factor very close to 1.0?

**Tags:** black-hole, gravity, neutron-star
> **Comments:**
> **username_1**: Are you sure about your formula? How did you derive it?  
> **username_2**: Your time dilation formula (for $1-t_0/t_f$) is an approximation. It's ok when $gr/c^2$ is small, but not when GR effects become significant. Instead, use $t_0 = t_f\sqrt{1 - \frac{r_s}{r}}$, where $r_s=2GM/c^2$, which can be found at https://en.wikipedia.org/wiki/Gravitational_time_dilation#Outside_a_non-rotating_sphere  
> **username_2**: Here's a Schwarzschild radius & time dilation [calculator](https://sagecell.sagemath.org/?z=eJxVjsFqwzAQRO_6ioFAbbVJI9kOjgyGntpTT7k7CEshAstOJZnifn3XKZRWoMPO7NuZDU799VOHr9hf3WAQtHFzxAOS8xbGDTq5aWQbvL1juuA0j3AjfFfuY1cwP5_QQpZFLYuqEqKSz6pU5RGPkKJTTM-rXamDqo-1qIVgPQmFUrUqqsORsRc3Jht0n5ixF5xz3-ZZnAYd4HWMNmZbSL5FINm4mPTYW-Sek1wKIcjRc5rO883oZNtXPUTLGwZ6Ia5B1MOvn2ru0VPh1boFCs2zcI4N3Qlxi8xn_G4tBBG5R7iPydAcP0LKJXZY-F88mQYgPpkfeV1dt5L5HfMFTzBdwengv2i5I5pYw78B52Bjtw==&lang=sage). You can put expressions in the input fields,  eg `10*au`  gives a distance of 10  AU.  
> **username_0**: @PM2Ring Schwartzchild got this wrong. Mass is decelerated away from the surface by gravity and therefore has an escape velocity. Light is accelerated away from the surface by gravity, and can therefore escape regardless of the surface gravity. This not so subtle difference between mass and light appears to have been quite readily ignored by Schwartzchild and many others since.  
> **username_1**: @AlanGee - Sorry, but this does not make much sense. Light is not "accelerated away from the surface": unless pointed radially it follows a curved null geodesic and as the Schwarzschild radius is approached most such geodesics end up captured by the surface. And "rate of length contraction factor" is an entirely nonstandard term that doesn't make much sense. Are you trying to measure the difference in radial geodesic distance from coordinate distance?  
> **username_0**: My understanding of gravity is based on the equation of free fall:- $dE/ds = -dp/dt$ that is applicable to both mass and light. For light, this means that the downward force generated by the energy gradient due to time dilation must be opposed by an equal and opposit upward force generated by a momentum gradient due to length contraction. Given that, it is easy to see that light is decelerated towards the surface, since its wavelength is reducing and its period is increasing in that direction. Of course if you measure it on any geodesic it will always look the same.  

---

## 1 answers

### Answer by username_3 on 2022-05-03. Score: 10

I'm not sure what you mean by gravitational length contraction, and the time dilation formula you are using is only approximately correct for weak gravitational fields.

Observed neutron stars are indeed only a small factor larger than their Schwarzschild radii. If they were just a little smaller (say $<1.4 r\_s$) then they would become unstable and likely collapse to become black holes.

Is this a coincidence? Well yes and no. It is the reason why we do not see neutron stars with masses of 3 times the sun or more. Such objects have indeed collapsed to have become black holes.

I suppose you could argue that it is an interesting twist of astrophysics that the massive stars that end up producing neutron stars finish their lives with iron cores that are too big to be supported by electron degeneracy pressure, but still small enough that they can avoid becoming a black hole. i.e. That the Chandrasekhar mass is smaller than the maximum mass for a neutron star. It wouldn't take much of a weakening of the strong nuclear force between closely packed neutrons to make forming neutron stars practically impossible. This is because neutron degeneracy pressure alone can only support neutron stars up to 0.75 solar masses and these could not be produced during stellar evolution without becoming stuck as stable white dwarfs. Anything more massive would then collapse directly to a black hole.

> **Comments:**
> **username_0**: Of course there is gravitational length contraction. That's why Einstein talked about space-time warping and not just time warping. It's also why a photon doesn't lose or gain the ability to do work when it passes through a gravitational field. I would need to read up on electron and neutron degeneracy forces to understand your answer better, but I certainly find it interesting.  
> **username_4**: High mass stars are supported by nuclear fusion tho, which neutron stars are not, so I'm not sure what the twist is.  
> **username_3**: @DaddyKropotkin the cores of high-mass stars at the ends of their lives are supported (briefly) by electron degeneracy pressure. It is an accident of nuclear physics that the Chandrasekhar mass is smaller than the maximum mass of a neutron star.  
> **username_4**: Oh I see, it was not clear to me that you meant at the end of nuclear fusion...  

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