Stars and Relativity by I.D. Novikov

Author:I.D. Novikov [Novikov, I.D.]
Language: eng
Format: epub
ISBN: 9780486171326
Barnesnoble:
Published: 2022-07-05T21:40:59+00:00

10.7 Relativistic Equations for Rotating Stars

ds2 - ( 1 - 2GM/rc2)dt2 + (4GK/c1r2)dtdq,

( 10. 7.3)

- [ 1 + O(GM/rc2)] [dr2 + r2d92 + r2 sin 2 9 dif>2] .

Here O(GM)/rc2) is a correction term which depends on the precise choice of coordinates. The key points are that (i) as for a nonrotating star, so also here, it is the Newtonian correction, - 2GM /rc2, to g11 that reveals the star’s total mass ; and (ii) as in the weak-field limit (§ 1 . 10) , so also here, it is the off-diagonal g,. term that reveals the total angular momentum.

The “fluid” inside the star is assumed to rotate in the !/>-direction. Thus, its four-velocity must have the form

( 10. 7.4)

If a distant observer with X-ray vision watches a particular fluid element at (x1, x2) , he sees it rotate with an angular velocity of (10.7.S )

(Recall that the proper time measured by his clock is the same as coordinate time t.) By using the normalization condition uiu; = 1 , we can reexpress the fluid ‘s four-velocity as ( 10.7.6)

Notice that rigid rotation corresponds to O(x1, x2) = constant. In that case the distant observer sees the star’s fluid to rotate as a solid body, with the distance between neighboring particles remaining forever constant.

Let us now turn to the energy considerations and variational principle that we promised.

We begin by calculating the change in the star’s total mass-energy M

when a unit amount of rest mass is added to it. We expect an answer which reduces to equation ( 10.6.22)-i.e. , l>M /l>Mo = (gu) 112-when the star is nonrotating and the mass is added at its surface.

We add the rest mass l>Mo by the following idealized process. An astrophysicist far from the star drops a bundle of rest mass l>Mo down an idealized pipe, which is inserted into the star, to a colleague who rides with the fiuid ring at (x1, x2) . The colleague catches the mass, inserts it into his fiuid ring, and then throws any energy that remains back up the pipe to the distant astrophysicist. The astrophysicist then uses conservation of mass-energy to determine the change in the star’s mass.

The energy balance for this injection process is this : 1 . The astrophysicist is careful to drop the bundle of rest mass l>Mo so that it has zero initial angular momentum and zero initial kinetic energy.

Thus, its initial four-momentum is

(10.7.7)

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