Speed of Light Paradox: Neutron Star Rotation Rate

[jflies]

TL;DR

If a neutron star spins fast enough, doesn't it imply that either (1) surface speeds greater than the speed of light are possible or (2) angular momentum is not conserved?

Sorry if this is a stupid question but I couldn't find an answer anywhere. According to 2 scientific papers, the neutron star PSR J1748-2446ad has a rotation rate of 716Hz, which equates to a linear surface speed of 0.24c. What if this star was originally rotating, let's say, 5 times (or more) faster before it collapsed. That would imply that it's surface speed would reach 1.2c. How could the conservation of momentum hold true unless speeds greater than 1c are allowed?

 

[Ibix]

Angular momentum does not scale linearly with angular velocity. It approaches infinity as the linear speed approaches $c$. So any initial angular momentum can be accommodated in the collapsed state without exceeding the speed of light.

Note that neutron stars are extremely complicated objects and you really need to think about their full stress-energy tensor.

 

[jflies]

Ok thanks for the quick response (and not making fun of my lack of knowledge)!

 

[PeroK]

Summary: If a neutron star spins fast enough, doesn't it imply that either (1) surface speeds greater than the speed of light are possible or (2) angular momentum is not conserved?

Sorry if this is a stupid question but I couldn't find an answer anywhere. According to 2 scientific papers, the neutron star PSR J1748-2446ad has a rotation rate of 716Hz, which equates to a linear surface speed of 0.24c. What if this star was originally rotating, let's say, 5 times (or more) faster before it collapsed. That would imply that it's surface speed would reach 1.2c. How could the conservation of momentum hold true unless speeds greater than 1c are allowed?

Momentum does not increase linearly with speed. Linear momentum, $p$, for example is given by:

$p = \frac{mv}{\sqrt{1-v^2/c^2}}$

You can see from this formula that any momentum can be obtained (no matter how large) with a speed of less than $c$.

When $v$ is small compared to $c$, then you get the classical, Newtonian approximation:

$p \approx mv$

But that only holds for speeds much less than $c$.

 

[jflies]

Momentum does not increase linearly with speed. Linear momentum, $p$, for example is given by:

$p = \frac{mv}{\sqrt{1-v^2/c^2}}$

You can see from this formula that any momentum can be obtained (no matter how large) with a speed of less than $c$.

When $v$ is small compared to $c$, then you get the classical, Newtonian approximation:

$p \approx mv$

But that only holds for speeds much less than $c$.

Yup, I was only considering the Newtonian approximation. Thanks for the clarification.