<p>The inferred radius by a distant observer is given by
<span class="math-container">$$ R_{\infty} = \frac{R}{\sqrt{1- R_s/R}},$$</span>
where <span class="math-container">$R_s = 2GM/c^2$</span>, <span class="math-container">$M$</span> is the gravitational mass and <span class="math-container">$R$</span> is the radius measured at the surface.</p>

<p>The fact that <span class="math-container">$R_{\infty} &gt; R$</span> is because an observer can see more than 50 per cent of the neutron star surface. See <a href="https://physics.stackexchange.com/questions/350805/seeing-something-from-only-one-angle-means-you-have-only-seen-what-of-its-su/350814#350814">https://physics.stackexchange.com/questions/350805/seeing-something-from-only-one-angle-means-you-have-only-seen-what-of-its-su/350814#350814</a></p>

CC BY-SA 4.0
2019-11-01T20:27:47.093Z
2019-11-01T20:27:47.093Z
ProfRob
<p>If we ignore the rotation issue (as Pela pointed out, the correction is &lt;1%), then we can approximate the spacetime curvature inside the neutron star using the <a href="https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric" rel="noreferrer">interior Schwarzschild metric</a> and use <a href="https://en.wikipedia.org/wiki/Exterior_Schwarzschild_metric" rel="noreferrer">the exterior one</a> for the outside. The <span class="math-container">$r$</span> coordinate is <em>not</em> the radial distance as one would expect, but defined in terms of <span class="math-container">$r=$</span> constant circles having circumference <span class="math-container">$2\pi r$</span>.</p>

<p>A neutron star of radius <span class="math-container">$R$</span> in these metrics has a circumference measured on its surface as <span class="math-container">$2\pi R$</span> as expected. But the distance to the core (that is, the length of a hole from the surface to the core) is: <span class="math-container">$$d(R) = \int_0^{R} \frac{dr}{1-Kr^2} =  \frac{1}{\sqrt{K}}\tanh^{-1}\left(\sqrt{K}R\right).$$</span>
where <span class="math-container">$K=r_s/R^3$</span> and <span class="math-container">$r_s$</span> is the Schwarzschild radius of the star (all of this assumes constant density in the interior). If we plot this for stars of different <span class="math-container">$R$</span> (but same mass, say one solar mass, producing a constant <span class="math-container">$r_s$</span>) we get the following:</p>

<p><a href="https://i.sstatic.net/6icBJ.png" rel="noreferrer"><img src="https://i.sstatic.net/6icBJ.png" alt="enter image description here"></a></p>

<p>As the neutron star approaches becoming a black hole it gets "deeper": there is more volume than one would expect. Less dense neutron stars have depths that scale linearly with their radius... except that it is <span class="math-container">$r_s/3$</span> larger! This odd result comes from the assumption that we look at stars of the same mass even though we make them much larger. A big constant mass object will curve spacetime too, and this produces this effect. (Yes, this means that the sun is <span class="math-container">$r_s/3 = 984.73$</span> meters deeper than it looks!)</p>

<p>If we instead decide that we use a constant density (say nuclear density) and plot the depth, the collapse to a black hole instead happens to the right as the neutronium sphere becomes too large. Here I still use the solar Schwarzschild radius as a length scale to keep things comparable. Now, for small spheres the coordinate and measured depths converge:</p>

<p><a href="https://i.sstatic.net/IGjjv.png" rel="noreferrer"><img src="https://i.sstatic.net/IGjjv.png" alt="enter image description here"></a></p>

<p>In actual neutron stars things are complicated by rotation, the core pressure diverging as one approaches <span class="math-container">$(9/8)r_s$</span>, and of course that it is hard to dig a hole through superfluid neutrons. </p>

CC BY-SA 4.0
2019-11-01T22:14:10.473Z
2019-11-01T22:14:10.473Z
Anders Sandberg
<p>What is the estimated size of neutron stars observed in their reference frame and in our reference frame?</p>

<p>That is, how bent is space-time around neutron stars?</p>

CC BY-SA 4.0
2019-11-01T12:42:10.140Z
This is actually a quite non-trivial question to answer, due to the so-called [Ehrenfest paradox](https://en.wikipedia.org/wiki/Ehrenfest_paradox) regarding Lorentz contraction of a rotating body, but since the involved velocities are, after all, small compared to the speed of light (at most ~10%), the difference which involves the factor $\sqrt{1-v^2/c^2}$ will be on the the <1% scale.
pela
<space-time><neutron-star>
Neutron star size in different reference frames
CoffeDeveloper
# Neutron star size in different reference frames

(Asked by: username_0 on 2019-11-01)

What is the estimated size of neutron stars observed in their reference frame and in our reference frame?

That is, how bent is space-time around neutron stars?

**Tags:** space-time, neutron-star
> **Comments:**
> **username_1**: This is actually a quite non-trivial question to answer, due to the so-called [Ehrenfest paradox](https://en.wikipedia.org/wiki/Ehrenfest_paradox) regarding Lorentz contraction of a rotating body, but since the involved velocities are, after all, small compared to the speed of light (at most ~10%), the difference which involves the factor $\sqrt{1-v^2/c^2}$ will be on the the <1% scale.  

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## 2 answers

### Answer by username_3 on 2019-11-01. Score: 5 | Accepted answer

If we ignore the rotation issue (as Pela pointed out, the correction is <1%), then we can approximate the spacetime curvature inside the neutron star using the [interior Schwarzschild metric](https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric) and use [the exterior one](https://en.wikipedia.org/wiki/Exterior_Schwarzschild_metric) for the outside. The $r$ coordinate is *not* the radial distance as one would expect, but defined in terms of $r=$ constant circles having circumference $2\pi r$.

A neutron star of radius $R$ in these metrics has a circumference measured on its surface as $2\pi R$ as expected. But the distance to the core (that is, the length of a hole from the surface to the core) is: $$d(R) = \int\_0^{R} \frac{dr}{1-Kr^2} = \frac{1}{\sqrt{K}}\tanh^{-1}\left(\sqrt{K}R\right).$$
where $K=r\_s/R^3$ and $r\_s$ is the Schwarzschild radius of the star (all of this assumes constant density in the interior). If we plot this for stars of different $R$ (but same mass, say one solar mass, producing a constant $r\_s$) we get the following:

[![enter image description here](https://i.sstatic.net/6icBJ.png)](https://i.sstatic.net/6icBJ.png)

As the neutron star approaches becoming a black hole it gets "deeper": there is more volume than one would expect. Less dense neutron stars have depths that scale linearly with their radius... except that it is $r\_s/3$ larger! This odd result comes from the assumption that we look at stars of the same mass even though we make them much larger. A big constant mass object will curve spacetime too, and this produces this effect. (Yes, this means that the sun is $r\_s/3 = 984.73$ meters deeper than it looks!)

If we instead decide that we use a constant density (say nuclear density) and plot the depth, the collapse to a black hole instead happens to the right as the neutronium sphere becomes too large. Here I still use the solar Schwarzschild radius as a length scale to keep things comparable. Now, for small spheres the coordinate and measured depths converge:

[![enter image description here](https://i.sstatic.net/IGjjv.png)](https://i.sstatic.net/IGjjv.png)

In actual neutron stars things are complicated by rotation, the core pressure diverging as one approaches $(9/8)r\_s$, and of course that it is hard to dig a hole through superfluid neutrons.


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### Answer by username_2 on 2019-11-01. Score: 2

The inferred radius by a distant observer is given by
$$ R\_{\infty} = \frac{R}{\sqrt{1- R\_s/R}},$$
where $R\_s = 2GM/c^2$, $M$ is the gravitational mass and $R$ is the radius measured at the surface.

The fact that $R\_{\infty} > R$ is because an observer can see more than 50 per cent of the neutron star surface. See <https://physics.stackexchange.com/questions/350805/seeing-something-from-only-one-angle-means-you-have-only-seen-what-of-its-su/350814#350814>


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