Null Killing Vector & Conserved Quantity - Physics Forums

[pervect]

If space-like Killing vectors are associated with a conserved momentum, and timelike Killing vectors are associated with a conserved energy, what is the conserved quantity associated with a null Killing vector?

For instance, "v" in ingoing Eddington Finklestein coordinates.
https://www.physicsforums.com/showthread.php?t=656805#post4637332

The line element is:

−(1−2m/r) dv^2+2 dv dr

None of the metric coefficient's are a function of v (and GR Tensor's KillingTest confirms that $\partial / \partial v$ is a Killing vector).

 

[George Jones]

If space-like Killing vectors are associated with a conserved momentum, and timelike Killing vectors are associated with a conserved energy, what is the conserved quantity associated with a null Killing vector?

For instance, "v" in ingoing Eddington Finklestein coordinates.
https://www.physicsforums.com/showthread.php?t=656805#post4637332

The line element is:

−(1−2m/r) dv^2+2 dv dr

None of the metric coefficient's are a function of v (and GR Tensor's KillingTest confirms that $\partial / \partial v$ is a Killing vector).

$\partial / \partial v$ is not lightlike.

A related post:

Unfortunately, there is some subtlety here, and this subtlety seems to have confused Hobson, Efstathiou, and Lasenby (HEL). Most of the subtlety has to do with Woodhouse's "second fundamental confusion of calculus."

By HEL's own definition on page 248,

... fix the other coordinates at their values at P and consider an infinitesimal variation $dx^\mu$ in the coordinate of interest. If the corresponding change in the interval $ds^2$ is positive, zero or negative, then $x^\mu$ is timelike, null or spacelike respectively.

$p$ in Eddington-FinkelStein coordinates $\left(p,r,\theta,\phi \right)$ is a timelike coordinate, not a null coordinate. To see this, apply HEL's prescription on page 248 to equation (11.6). Varing $p$ while holding $r$, $\theta$, and $\phi$ constant gives $dr = d\theta = d\phi = 0$ and

$$ds^2 = \left( 1 - \frac{2M}{r} \right) dp^2.$$

Hence, (when $r > 2M$) $ds^2$ is positive, and $p$ is a timelike coordinate.

HEL are thinking of $p$ in Kruskal coordinates $\left(p,q,\theta,\phi \right).$. In this case, applying the page 248 prescription to equation (11.16) gives that $p$ is a null coordinate. Do you see why?

What type of coordinate is $r$ in Eddington-FinkelStein coordinates $\left(p,r,\theta,\phi \right)$?

By now, you should be thoroughly confused! How can the "same" $p$ be timelike in one set of coordinates and null in another set of coordinates? If you want, I am willing to spend some time explaining in detail what is going on here, and what Woodhouse's "second fundamental confusion of calculus" is.

 

[PeterDonis]

$\partial / \partial v$ is not lightlike.

Except at $r = 2M$, correct? If I'm reading this right, the ingoing E-F $v$ coordinate is timelike for $r > 2M$, null at $r = 2M$, and spacelike for $r < 2M$, just like the Painleve $T$ coordinate.

Also, for extra fun, if I'm reading this right, since ingoing radial null geodesics are curves of constant $v$, in ingoing E-F coordinates, $\partial / \partial r$ is null! (Whereas in Painleve coordinates, $\partial / \partial r$ is spacelike, and in Schwarzschild coordinates, $\partial / \partial r$ is spacelike for $r > 2M$, null at $r = 2M$, and timelike for $r < 2M$.)

I think recognizing all this also puts the question in the OP in a new light. Consider, for example, $\partial / \partial T$ in Painleve coordinates (or $\partial / \partial t$ in Schwarzschild coordinates, since the integral curves of both are the same). This is a KVF, and it is timelike outside the horizon, null on the horizon, and spacelike inside the horizon. The constant of motion associated with it is usually called "energy at infinity". Of course that interpretation is usually considered to be problematic on or inside the horizon; but does that mean that the same constant of the motion (which is constant all along an infalling geodesic, so the energy at infinity of an infalling object is the same all the way from infinity down to the singularity at $r = 0$) somehow becomes a "momentum" inside the horizon because the associated KVF is now spacelike?

 

[pervect]

Ah-ha! Thanks, George - I'm familiar with the second fundamental confusion you describe, but apparently not familiar enough to avoid falling victim to it again.