CPT (M?) symmetries in Kerr-Newman metric

[PeterDonis]

What exactly does time reversal entail? And how do you represent it mathematically??

I think at this point we need to go back to basics. First, let's restrict attention to what you called "local" transformations; i.e., we're looking at a specific event in spacetime and what happens when we do various transformations on the local coordinates. But we can always write local coordinates in Minkowski form, so what we are really looking at with the local transformations is transformations on Minkowski spacetime--more precisely, we are looking at *isometries* of Minkowski spacetime, transformations that leave the Minkowski metric invariant.

The full set of such transformations is the 10-parameter Poincare group:

http://en.wikipedia.org/wiki/Poincaré_group

The 10 parameters break up into 4 translations (one in each of the four spacetime dimensions) and 6 "rotations", and the latter, if you choose a particular local inertial frame, can be further broken up into 3 spatial rotations and 3 boosts.

However, there's also another way of looking at this group, as the Wikipedia article states:

"As a topological space, the group has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time reversed and spatially inverted."

In other words, there are four transformations we can pick out of the whole Poincare group that have special significance: 1 (the identity), P (parity), T (time reversal), and PT (combined parity and time reversal). These four are special because they form a sort of "basis" of the group: any arbitrary element of the Poincare group can be expressed as the composition of one of these four transformations with a transformation from the restricted Lorentz group, the group of "Lorentz transformations" that is usually talked about in basic special relativity courses (this restricted group is really just the component of the Poincare group that is connected to the identity).

So to find a suitable mathematical expression for time reversal, we need to first pick a representation of the Poincare group to work with. But since the full Poincare group is basically just four "copies" of the Lorentz group, each multiplied by one of (1, P, T, PT), we really just need to pick a representation of the Lorentz group:

http://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group

Then we need to make sure that this representation still works when we extend it to cover the full Poincare group by multiplying by (1, P, T, PT), with suitable representations for those four transformations. Doing that last step should give us a mathematical representation of time reversal in a local inertial frame. We could then look at how to transform that to a global set of coordinates.

Of course the problem with the above is that there are lots of representations! However, as you can see from the Wiki article I just linked to, which representation you want to use is really dictated by what you are trying to represent, e.g., if you want to represent the electromagnetic 4-potential you use the (1/2, 1/2) or 4-vector representation. So when we get to the point of looking at how time reversal acts on a complicated expression like that for the Kerr metric, we may actually be dealing with compositions of objects from different representations.

Having just thrown a whole bunch of additional complexity into the mix , I think it's best to stop for now and take some time to digest.

Edit: Saw your edit after posting, the above may be moot but I think it's still useful background. Will wait to see your further post.

 

[michael879]

Ok I've stepped through this pretty rigourously, and I'm confident of, although confused by, my results. First off, the source I've been using to get the metric is http://en.wikipedia.org/wiki/Kerr%E2%80%93Newman_metric. I've also been using a general relativity book I have to brush up on some things.

Now for the analysis: the line element is given by d\tau^2 = g_{\mu\nu}dx^\mu dx^\nu. d\tau^2 is a Lorentz scalar, and so it should remain unchanged under any Lorentz transformations. Since dx^\mu is a "true" 4-vector ("true" in the sense that it transforms as a 4-vector for any Lorentz transformation without any extra sign changes), g_{\mu\nu} must be a "true" rank 2 4-tensor. The Kerr-Newman metric in Kerr-Schild form is given by:

g_{\mu\nu} = \eta_{\mu\nu} + fk_\mu k_\nu

given the transformation properties of g and η, f must be a Lorentz scalar, and k must be a 4-vector or a 4-pseudovector (or any combination of the two). These two quantities are defined as
f = \dfrac{Gr^2}{r^4+a^2z^2}\left(2Mr-Q^2\right)
k_t = 1
k_x = \dfrac{rx+ay}{r^2+a^2}
k_y = \dfrac{ry-ax}{r^2+a^2}
k_z = \dfrac{z}{r}
and a is the usual J/M.

The 4-potential (which is not a "true" 4-vector because it is a pseudo vector under time-reversal) is defined as
A_\mu = \dfrac{Qr^3}{r^4+a^2z^2}k_\mu

Now that I've setup the metric, I want to talk about 3 transformations that I will call P, R, and T.

P: x→-x, y→-y, z→-z
=================
Looking at the above equations, it is clear that k (and therefore A) transforms as a 4-vector, and f remains constant. Therefore the symmetry requires are satisfied and parity is a trivial symmetry of the Kerr-Newman black hole.

R: r→-r
=================
Although this is not a Lorentz transformations (so no symmetry arguments can be made), it is particularly interesting. After performing the transformation, you can reorganize the minus signs to get:
f = \dfrac{Gr^2}{r^4+(-a)^2(-z)^2}\left(2(-M)r-(-Q)^2\right)
k_t = 1
k_x = \dfrac{r(-x)+(-a)(-y)}{r^2+(-a)^2}
k_y = \dfrac{r(-y)-(-a)(-x)}{r^2+(-a)^2}
k_z = \dfrac{(-z)}{r}
A_\mu = \dfrac{(-Q)r^3}{r^4+(-a)^2(-z)^2}k_\mu
It now becomes clear what the black hole would "look" like after an R transformation. R[M,J,Q] \equiv P[-M,J,-Q] (not equal, but identical fields), i.e. an R transformation produces a metric which is identical to the original under a parity transformation with opposite signed mass and charge. However, this was just some manipulation of the equations, and the actual black hole still has inertial mass M, and electric charge Q.

What's interesting about this though, is that it connects R and P transformations, making R similar to some kind of parity transformation. Additionally, since the sign of r only affects the black hole, the R transformation is a transformation of some kind of intrinsic property of the black hole. It basically "flips" the black hole between the two alternate universes. I never understood the concept of chirality in QFT intuitively, but the strangeness of this reminds me of it a lot..

T: t→-t
=================
It is this transformation that has me completely confused. The only part of the metric that depends explicitly on time is the a parameter (because J→-J). This alone does not transform the metric even close to correctly. So there are two options:
1) time reversal is not a symmetry of general relativity. However, time reversal is an improper Lorentz transformation and therefore should be a symmetry.
2) there are other values that transform under time reversal that aren't immediately obvious.

After looking through the metric, I found that there is only one way to make it transform correctly. If t→-t implies r→-r and M→-M, f transforms correctly as a scalar, and k transforms as a pseudo 4-vector. Note, that in this case the actual mass must change sign, and not just the active gravitational mass (because changing the sign of inertial mass changes the sign of J, leaving a with an overall minus sign still). In addition to enforcing the metric transformation, the 4-potential also needs to transform as a pseudo-vector. The dependence on r^3 though, makes it transform as a 4-vector (the scalar potential changes sign, and the vector potential doesnt). Therefore t→-t must also make Q→-Q for the electromagnetic fields to transform properly.

So the result is that there are two possibilities. Either d\tau^2 is not a true Lorentz scalar (which I'm pretty sure would invalidate General Relativity), or T[M,J,Q] = R[-M, J, -Q] (i.e. t→-t makes r→-r, M→-M, Q→-Q, and J→-(-J)=J). Another interesting feature is that using the above analysis for R transformations, T[M,J,Q] \equiv P[M,J,Q] (again, not equal because T[M,J,Q] has negative inertial mass and negative electric charge). Although this should be obvious from the transformation properties.

 

[PeterDonis]

R: r→-r

I understand what this transformation is doing mathematically, but I'm not sure it has any actual physical meaning.

T: t→-t
=================
It is this transformation that has me completely confused. The only part of the metric that depends explicitly on time is the a parameter (because J→-J). This alone does not transform the metric even close to correctly.

What would you consider "correctly"? As I read it, J -> -J (or equivalently a -> -a) converts (loosely speaking) a black hole rotating clockwise into a black hole rotating counterclockwise, with no other parameters changed. This seems OK to me after thinking it over, since the same would be true for an ordinary object. See further comments below.

So there are two options:
1) time reversal is not a symmetry of general relativity. However, time reversal is an improper Lorentz transformation and therefore should be a symmetry.

After thinking about this quite a bit, I think we've been confusing symmetry of a *theory* with symmetry of particular *solutions*.

For example: one could say that special relativity "has time reversal symmetry", but one can interpret that statement in two ways, which both happen to be true in that particular case. One could say that the general *equations* of special relativity are symmetric under time reversal; but one could also say that *Minkowski spacetime*, which happens to be the unique spacetime that satisfies the general equations of SR, is symmetric under time reversal. More precisely, your transformation T is an isometry of Minkowski spacetime; it leaves the metric invariant.

I think what we're seeing with Kerr-Newman spacetime is that these two notions become distinct. The general equation of GR, the Einstein Field Equation, of which the Kerr-Newman spacetimes constitute just one family of solutions, is "time reversal symmetric"; but what that means is not that nothing can change under time reversal, but that the time reverse of any solution of the EFE is also a solution of the EFE.

If a solution's time reverse happens to be that same identical solution (as in the case of Minkowski spacetime, or if that's too simple, take the case of Schwarzschild spacetime, which goes into itself under time reversal), then the individual solution is "time reversal symmetric"--again, your transformation T is an isometry of that spacetime.

But in the case of Kerr-Newman, the time reverse of a given solution is not the same solution, but a solution which looks the same except for the reversal of sign of the angular momentum. So the time reversal symmetry of the underlying theory is manifested by solutions occurring in pairs which are the time reverses of each other (except for the case a = 0, which reduces to Schwarzschild and therefore time reverses into itself--I am leaving out the case of nonzero electric charge here).

 

[michael879]

I get what your saying, but isn't one of the postulates of GR basically that the line element is a Lorentz scalar??

This means under any Lorentz transformation it stays the same (i.e. the infinitesimal proper length is invariant of reference frame).

The only way for this to be true is for the spacetime metric to transform as a tensor under any Lorentz transformation.

If the metric doesn't transform as a tensor under some transformation, then the line element will not transform as a scalar.

Time reversal T is an inproper Lorentz transformation (detT = -1).

The Kerr-Newman metric does not transform as a tensor under T if you only take into account the transformation of a. This means that under time reversal the line element changes, violating the initial postulate.

What I'm saying is this deeper symmetry constraint on the metric is what causes the EF equations to be symmetric under Lorentz transformations, not the other way around. Where is the flaw in the above logic?

Also, how would you explain why the 4-potential of a KN black hole does not transform correctly under time reversal? It is clear from Maxwells equations that under time reversal A→-A and \phi→\phi. However the metric and 4-potential of the KN black hole are related by the 4-vector k. k does not transform as a vector by simply setting a→-a under time reversal, and therefore the 4-potential doesn't transform as a vector...

 

[PeterDonis]

I get what your saying, but isn't one of the postulates of GR basically that the line element is a Lorentz scalar??

Yes, but you have to be careful in specifying exactly what that means.

To lead into what I'm getting at, in going back over the thread I realized that I missed something in my last post. I posted earlier that time reversal changes the sign of J (or a), but also changes the sign of dt (but not dphi). If that's correct, then the dt dphi term of the metric would *not* change sign under time reversal; J changes sign but so does dt, so the two sign changes cancel.

But what does this actually mean, physically? Since dt and dphi are the critical differentials here, consider a small line element where only those two differentials are nonzero. In the "forward time" case, we have a scalar composed of three terms: a dt^2 term, a dphi^2 term, and a dt dphi term (which has a factor of J in it).

What happens when we do a time reversal? Think of the line element as a small line segment, which in the "forward time" case went from (t, phi) to (t + dt, phi + dphi). Under time reversal, it now goes from (t', phi) to (t' - dt', phi + dphi), because the "direction" of the time coordinate has flipped. So the length of the actual physical line element does not change--what happens is that the same physical line element is now described by a slighly different set of coordinates due to the time reversal.

What makes this look a bit confusing is that if you just look at the formula for the line element, without considering exactly what the coordinate differentials mean physically, it *looks* as though the line element has changed, because the sign of the dt dphi term has indeed flipped in the formula. But when you actually look at how the formula *describes* the same actual, physical line element, you see that there's a sign change in dt as well, which results in the same final scalar describing the same physical line element.

 

[michael879]

I totally agree with what you just said, but I'm talking about the Kerr-Schild form of the metric not the Boyer-Linquist one. I have no idea what the 4-potential is in the Boyer-Linquist formulism, which is why I've been sticking to the Kerr-Schild form. In the Boyer-Linquist form, time reversal changes the sign of a and dt, which cancel out exactly, preserving the line element. The metric has 1 off diagonal term in phi-time. This component has an overall factor of a, so that it changes sign under time reversal.

Therefore the metric, and the line element transform correctly in the Boyer-Linquist form. However I don't know the form of the 4-potential, so I can't say if that transforms correctly as well.

In the Kerr-Schild form, the line element changes under time reversal (if you just take into account the sign changes of a and dt), and the metric and 4-potential do not transform correctly. The only way to rectify this, that I can see, is to either say the Kerr-Schild form is wrong, or to make the additional transformations I mentioned above. Note, that in the Boyer-Linquist form the additional transformations do not change the line element either, so that everything still transforms correctly. I would be interested to see what the 4-potential is though, because I would suspect it could only transform correctly given the additional transformations (and for whatever reason the Boyer-Linquist form hides them).

 

[michael879]

Also, I just noticed you're over complicating the situation. The line element is given simply by
ds^2 = g_{\mu\nu}dx^\mu dx^\nu
The "metric terms" are contained in g, and dx is a 4-vector. Both g and dx will change under Lorentz transformations, but the changes will cancel out to leave ds^2 unchanged. It is trivial to show that the Boyer-Linquist line element and metric transform correctly under time reversal.

*edit* Also, none of the terms in the line element equation will change sign. You had the right idea but you made a few sign mistakes.

 

[PeterDonis]

I totally agree with what you just said, but I'm talking about the Kerr-Schild form of the metric not the Boyer-Linquist one.

But the property of time reversal invariance is independent of the form of the metric. Changing the form of the metric just corresponds to changing coordinate charts, which does not affect any physical quantities. So if two different forms of the metric appear to be giving two different answers, we must be doing something wrong. I'll take another look at the Kerr-Schild form and work through the computation in more detail.

Also, I just noticed you're over complicating the situation. The line element is given simply by
ds^2 = g_{\mu\nu}dx^\mu dx^\nu
The "metric terms" are contained in g, and dx is a 4-vector. Both g and dx will change under Lorentz transformations, but the changes will cancel out to leave ds^2 unchanged.

Yes, I know; that was my point, that you have to look at sign changes in both g *and* dx to show time reversal invariance. Just looking at the formula for g is not enough. See below.

*edit* Also, none of the terms in the line element equation will change sign. You had the right idea but you made a few sign mistakes.

I may have; but again, my general point is that "none of the terms in the line element equation will change sign" is true *only* if you correctly analyze the correspondence between specific line element expressions (i.e., with specific values inserted for the coordinate differentials) and specific physical line elements. If you just look at the general *formula* for the line element, with coordinate differentials not assigned any specific values, formally the sign of the J dt dphi term *does* change under time reversal, because you have changed coordinate charts; the line element equation with the sign of J flipped refers to a different chart than the original equation did, one in which the direction of the time axis is reversed. But a given *physical* line element will have the same actual number for ds^2 as it had before, because the coordinate differentials describing it will change--specifically, the sign of dt that describes that physical line element will flip.

 

[michael879]

But the property of time reversal invariance is independent of the form of the metric. Changing the form of the metric just corresponds to changing coordinate charts, which does not affect any physical quantities.

Well that was kindof my point. Let's just assume time reversal causes R, M, and Q sign changes. This would make the Kerr-Schild form of the metric transform correctly, and keep the line element invariant. It would also work for the Boyer-Lindquist form. The Boyer-Lindquist form doesn't disagree with the Kerr-Schild form if you make the above assumption, it just doesn't enforce that assumption (i.e. it doesn't prove or disprove it). And if the Boyer-Lindquist form is so clearly time reversal invariant, the Kerr-Schild form needs to be as well (which forces the RMQ changes).

 

[PeterDonis]

Lets just assume time reversal causes R, M, and Q sign changes...And if the Boyer-Lindquist form is so clearly time reversal invariant, the Kerr-Schild form needs to be as well (which forces the RMQ changes).

No, I don't agree. As I've said before, I don't think these additional sign changes have physical meaning. I think what's actually going on is that the Kerr-Schild form of the metric has two forms, "ingoing" and "outgoing", which are not identical; instead, one is the time reverse of the other. This is different from the Boyer-Lindquist form, which is its own time reverse.

Let me expand on that some more. An excellent reference for Kerr-Newman spacetime is Matt Visser's article here:

http://arxiv.org/abs/0706.0622

He describes the Kerr-Schild form in "spherical" coordinates first, which makes it easier to understand what's going on (at least for me). As you can see from his article, there are indeed two forms of the metric in these coordinates: an "ingoing" form and an "outgoing" form. This is similar to Eddington-Finkelstein or Painleve coordinates for Schwarzschild spacetime. And just as with those coordinates, the "ingoing" form, when time reversed, gives the "outgoing" form, and vice versa. So the sign changes you are seeing on time reversal are *supposed* to be there.

Let's go back to Painleve coordinates on Schwarzschild spacetime as an easier example to see what's actually going on. I think we established in earlier posts that the time reverse of the standard "ingoing" Painleve coordinates, which have a - dt dr term as the only non-diagonal term, is a line element with a + dt dr term (because the sign of dt flips), which is what you see in "outgoing" Painleve coordinates. So this line element expression *does* change under time reversal, *even* when we take into account the sign flips in both the form of g and in the coordinate differentials themselves (and I agree that we have to do that to see what happens to an actual physical line element).

Now suppose we thought this sign flip in the dt dr term was "wrong", and tried to "fix" it by changing the signs of M and/or r. Changing the sign of M alone, which would "fix" the sign of the dt dr term, won't work, because M also appears in the dt^2 term, and switching M's sign does *not* leave that term invariant. The same goes for changing the sign of r alone. And switching the sign of *both* M and r, which would leave the dt^2 term invariant, doesn't work either, because that would make the sign of the dt dr term flip again (the sign flips in M and r cancel and we're left with the sign flip in dt)! So trying to "fix" things by flipping the signs of M and/or r won't work (and that's OK, since as I've said, those sign flips don't make sense physically anyway).

Instead, we have to recognize that in Painleve coordinates, unlike in Schwarzschild coordinates, the surfaces of "constant time" are *not* left invariant by time reversal! Put another way, in Schwarzschild spacetime, the surfaces of constant Schwarzschild time t are taken into themselves by the time reversal transformation. But the surfaces of constant Painleve time are *not*; rather, time reversal takes the surfaces of constant "ingoing" Painleve time into surfaces of constant "outgoing" Painleve time (which are *different* surfaces), and vice versa. And *that* means that a line element which is, say, "pure" dr in "ingoing" Painleve coordinates (i.e., it lies completely within a single surface of constant ingoing time) will *not* be "pure" dr in "outgoing" Painleve coordinates--it will have a dt component as well. And vice versa.

So with Painleve coordinates, when figuring out the impact of time reversal, it's not enough just to look at the sign flips in dt and other things; you also have to take into account that coordinate differentials which were zero before may be nonzero after (or vice versa). The same thing applies to Kerr-Schild coordinates on Kerr-Newman spacetime, as compared to Boyer-Lindquist coordinates; time reversal leaves the surfaces of constant Boyer-Lindquist time invariant, but it does *not* leave the surfaces of constant "ingoing" Kerr-Schild time invariant. So all the sign changes must cancel in the Boyer-Lindquist form to leave the ds^2 of an actual, physical line element invariant (and they do), but they should *not* cancel in the Kerr-Schild form (and they don't).

 

[michael879]

ok I'm still not really satisfied with just accepting that the line element, which should be a Lorentz scalar, just changes for these coordinate systems... However, ignoring that for a second, how do you explain the 4-potential issue? The 4-potential is a "physical" object with well defined transformation properties. Why would time reversal completely change the 4-potential?

 

[PeterDonis]

ok I'm still not really satisfied with just accepting that the line element, which should be a Lorentz scalar, just changes for these coordinate systems...

Once again, the actual, physical length of a given actual, physical line element does *not* change. That's what "Lorentz scalar" means. But the exact *description* of a given actual, physical line element in terms of coordinate differentials *does* change, because time reversal means changing coordinate charts, and any change of coordinate charts changes the description of a given actual, physical line element in terms of coordinate differentials. What was confusing us was the assumption that the *only* change "time reversal" can make to the coordinate differentials is to flip the sign of dt; but in fact, for coordinates like Painleve or Kerr-Schild, that is *not* the case.

So to obtain the same actual number for ds^2 for the same actual, physical line element, the functional form of the line element in terms of the coordinate differentials has to change in other ways besides just what is needed to balance the sign change in dt. We simply haven't computed exactly what those other changes are. For example, I didn't compute what *other* coordinate differentials would change upon "time reversal" of Painleve coordinates; I simply showed that flipping the sign of dt alone does not leave ds^2 invariant, and no combination of sign flips of M and r can fix that. So obviously other differentials must change as well to keep ds^2 invariant.

The simplest case would be the one I described in my last post: a line element that is purely radial (only dr nonzero) in "ingoing" Painleve coordinates. Time reversal would leave dr the same (since the surfaces of constant r, the 2-spheres, *are* left invariant), but would add a nonzero dt; so there would now be nonzero dt^2 and dt dr terms in the formula for ds^2 in "outgoing" Painleve coordinates, in addition to the nonzero dr^2 term. The signs of these two new terms are opposite, and they should cancel each other, leaving the (unchanged) dr^2 term as the value of ds^2.

The 4-potential is a "physical" object with well defined transformation properties. Why would time reversal completely change the 4-potential?

It would change the *components* of the 4-potential in the new coordinate chart, as compared to the old. It would not change the actual, physical 4-vector that those components describe.

 

[PeterDonis]

I think we established in earlier posts that the time reverse of the standard "ingoing" Painleve coordinates, which have a - dt dr term as the only non-diagonal term, is a line element with a + dt dr term (because the sign of dt flips), which is what you see in "outgoing" Painleve coordinates.

I should clarify here that this is using the (+---) sign convention, which is the one used in the Wikipedia article on Painleve coordinates.

 

[michael879]

*sigh* I finally get it, what a let down.

The "dt" coordinate I've been using is actually \bar{dt} = dt - \dfrac{2Mr}{r^2-2Mr+a^2}dr. waaay too much algebra for me to work out, but I'm willing to bet that under a real time reversal, everything will transform correctly.

*edit* that equation alone explains why I was forced to flip the sign of M and r. If \bar{dt}\rightarrow-\bar{dt} and you only want to get dt\rightarrow-dt, you must flip M and r.

 

[michael879]

Ok, so given this new information, let's do a real time reversal: dt→-dt

d\bar{t} \rightarrow -dt - \dfrac{2(-M)(-r)}{(-r)^2-2(-M)(-r)+a^2}(-dr)\rightarrow-d\bar{t}|_{M\rightarrow-M,r\rightarrow-r}

f \rightarrow \dfrac{G(-r)^2}{(-r)^4+a^2z^2}\left(2(-M)(-r)-(-Q)^2\right)
k_\bar{t} \rightarrow 1
k_x \rightarrow -\dfrac{(-r)x+ay}{(-r)^2+a^2}
k_y \rightarrow -\dfrac{(-r)y-ax}{(-r)^2+a^2}
k_z \rightarrow -\dfrac{z}{(-r)}
A_\mu \rightarrow \dfrac{(-Q)(-r)^3}{(-r)^4+a^2z^2}k_\mu

so therefore,
g_{\mu\nu} \rightarrow \bar{T}[g_{\mu\nu}]|_{M\rightarrow-M,Q\rightarrow-Q,r\rightarrow-r, J\rightarrow J}
and
ds^2 = g_{\mu\nu}dx^\mu dx^\nu \rightarrow g_{\mu\nu}dx^\mu dx^\nu|_{M\rightarrow-M,Q\rightarrow-Q, J\rightarrow J,r\rightarrow-r}

Now assuming that the line element for [M,J,Q,r] is the same as that for \bar{T}[-M, J, -Q, -r], the line element is invariant under time reversal. And it turns out that it is, so although its far from obvious, the Kerr-Newman space-time is time reversal invariant! This is somewhat disappointing considering my original goal of showing a CPT symmetry in the Kerr-Newman geometry (which should be the geometry of fundamental particles). It seems to be trivial now though, and there is no CP violation.

 

[PeterDonis]

The "dt" coordinate I've been using is actually \bar{dt} = dt - \dfrac{2Mr}{r^2-2Mr+a^2}dr

For the Kerr-Schild form of the metric, yes, the "time" coordinate is not the same as the "t" coordinate in Boyer-Lindquist coordinates. See below for more discussion of this.

Ok, so given this new information, let's do a real time reversal:

I still don't understand why you're flipping the signs of M and r (or Q either, for that matter). Let me work through this in Painleve coordinates on Schwarzschild spacetime to illustrate how I'm looking at this.

Consider a line element where only dT and dR (I'll capitalize the T and R to make it clear that I mean Painleve T and R, *not* Schwarzschild T and R) are nonzero. I will also write "v" for the "escape velocity", \sqrt{2M / r}[/tex] to make the formulas easier to read. So in the &quot;ingoing&quot; Painleve coordinates we have (using the +--- sign convention I used earlier):<br /> <br /> ds^{2} = \left( 1 - v^{2} \right) dT^{2} - 2 v dT dR - dR^{2}<br /> <br /> Now, to make the signs of each term absolutely clear, let&#039;s put in absolute value signs and make the sign of dT and dR explicit. Suppose we are talking about the line element along the worldline of a &quot;Painleve observer&quot; who is free-falling into the black hole. Then we have dT = |dT| and dR = - |dR| = - |v| |dT|, where I have also put absolute value signs around v to make it clear that we are talking about the *magnitude* of v only. So we have the actual *number* ds^2 for this physical line element given by:<br /> <br /> ds^{2} = \left( 1 - v^{2} \right) dT^{2} - 2 |v| |dT| (- |v| |dT|) - v^{2} dT^{2} = dT^{2} \left(1 - v^{2} + 2 v^{2} - v^{2} \right) = dT^{2}<br /> <br /> Now this actual number has to be invariant under time reversal. What does the line element look like in the time reversed coordinate chart T&#039;, R&#039;? It looks like this: dT&#039; = - dT, dR&#039; = dR. In other words: dT&#039; = - |dT|, dR&#039; = - |v| |dT|. But that means that, to keep the actual number ds^2 invariant for the same physical line element, the line element equation in T&#039;, R&#039; must look like this:<br /> <br /> ds^{2} = \left( 1 - v^{2} \right) dT&amp;#039;^{2} + 2 v dT&amp;#039; dR&amp;#039; - dR&amp;#039;^{2}<br /> <br /> The sign of the second term must flip to compensate for the fact that dT&#039; = - dT. Substituting dT&#039; = - |dT| and dR&#039; = - |v| |dT| into the above line element will give the same result, ds^{2} = dT^{2}, but *only* with the sign flip in the second term of the above equation. In other words, the time reverse of ingoing Painleve coordinates is outgoing Painleve coordinates. And as I said before, flipping the signs of M and/or r will *not* keep ds^2 invariant under time reversal (because flipping the sign of either one individually flips the sign of *both* v and v^2, which does not leave ds^2 invariant, and flipping the sign of both leaves both v and v^2 invariant, so it does nothing). You *have* to change the sign of the second term in the line element equation.<br /> <br /> Now, what about the fact that the Painleve time coordinate T is not the same as the Schwarzschild time coordinate t? The Wikipedia page gives this (using my notation):<br /> <br /> dT = dt - \frac{v}{1 - v^{2}} dr<br /> <br /> Time reversing this, which is expressed as dt&#039; = - dt and dr&#039; = dr, gives:<br /> <br /> dT&amp;#039; = dt&amp;#039; + \frac{v}{1 - v^{2}} dr&amp;#039;<br /> <br /> Notice that the sign of dT&#039; is still the same as the sign of dt&#039;; the only change is the sign of the dr term. And since dr = dR (the radial coordinate is unchanged in going from the Schwarzschild chart to the Painleve chart), this shows *why* the sign of the second term in the Painleve line element formula above changes sign under time reversal (whereas the Schwarzschild line element formula is unchanged).<br /> <br /> <blockquote data-attributes="" data-quote="michael879" data-source="post: 3839366" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> michael879 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> And it turns out that it is, so although its far from obvious, the Kerr-Newman space-time is time reversal invariant! </div> </div> </blockquote><br /> Still not sure about this because of the sign change in J; as I said before, the time reverse of an object rotating counterclockwise should be an object rotating clockwise (speaking somewhat loosely), which is also a valid solution of the applicable equations, but not the *same* solution.

 

[michael879]

I still don't understand why you're flipping the signs of M and r (or Q either, for that matter).

This time I'm not, I was merely using an algebraic trick to show that
T[M,J,Q,r] \equiv \bar{T}[-M,-J,-Q,-r]
which can easily be shown to leave the line element unchanged. I made some minor mistakes regarding J, but I think my conclusion was correct.

dT&#039; = dt&#039; + \frac{v}{1 - v^{2}} dr&#039;

this is the main difference between the Painleve metric and the Kerr-Schild metric. The "time" coordinate in the Painleve metric changes sign under time reversal. Time reversal on the Kerr-Schild metric does not change the sign of the "time" coordinate, because the second term is just a function of M and r. If you want to make time reversal change the sign of the time coordinate, you must make M and r negative. I now understand that this is merely an algebraic trick to relate the two time coordinates, and not a physical phenomena

Still not sure about this because of the sign change in J; as I said before, the time reverse of an object rotating counterclockwise should be an object rotating clockwise (speaking somewhat loosely), which is also a valid solution of the applicable equations, but not the *same* solution.

Yes the time reverse of an object rotating counter clockwise IS an object rotating clockwise. However any Lorentz invariants regarding that object should not change under time reversal. Classical invariants of such an object are M, Q, \vec{J}^2, etc. The line element is just another Lorentz invariant and should never change under a Lorentz transformation. The metric, however, does change and becomes the metric of a Kerr-Newman black hole spinning in the opposite direction.

 

[PeterDonis]

Looks like we've finally got this one nailed. The only thing I still wanted to comment on is this:

the Kerr-Newman geometry (which should be the geometry of fundamental particles)

Only if fundamental particles are actually black holes. The problem with that hypothesis is that all known fundamental particles have masses well below the Planck mass, and although we don't have a complete theory of quantum gravity yet, what we do know about quantum gravity strongly suggests, AFAIK, that the smallest mass a black hole can have is the Planck mass.

I believe at least one physicist (can't remember who) has speculated that what we call "fundamental particles" are really just bound states of the fundamental entities of some underlying theory (perhaps string theory), while what we call "black holes" are just free states of the same fundamental entities. The motivation for this is that "fundamental particles" have a discrete mass spectrum, whereas "black holes" have a continuous mass spectrum. I don't know if any attempt has ever been made to expand on this theoretically.

 

[michael879]

The Planck mass thing is just some order of magnitude thing I believe. Below the Planck mass is when quantum effects of gravity should kick in. However there are some striking similarities between fundamental particles and black holes. The most surprising is probably the g=2 factor of both. There's also the no hair theorem, and some others, but what got me interested was the fact that every fundamental particle, if it were a black hole, would be naked. When you introduce naked singularities you get all kinds of strange stuff like nonlocality, nondeterminism, and strange causal behavior. However these three things are fundamental concepts in quantum, and bell's inequality actually proves that no interpretation can escape all three.

Regardless of whether or not particles are black holes, its hard to argue with the similarities they have. I was just trying to investigate the CP symmetry of QFT that takes a particle to its anti-particle. I'm actually kind-of leaning towards calling my R transformation a CP transformation, and my original T' reversal (which is not a time reversal) a T transformation. At distances far away from the black hole, R = CP and T' = T. The black hole would have a CPT symmetry, but applying CP or T would give it negative charge, negative parity, and negative active gravitational mass.

*edit* also, I know that the mass/charge/angular momentum are all discretized for fundamental particles, but charge and angular momentum are related by magnetic monopoles, and I was hoping to find some stability constraint on naked singularities that might give an additional requirement to constrain mass.

 

[PeterDonis]

The Planck mass thing is just some order of magnitude thing I believe. Below the Planck mass is when quantum effects of gravity should kick in.

Not necessarily *below* the Planck mass, but around that order of magnitude of spacetime curvature--that condition is more easily expressed as radius of curvature approximately equal to the Planck length, or energy density equal to one Planck mass per Planck volume (Planck length cubed).

However, that wasn't quite what I was referring to. I was referring to the fact that, as we currently understand black holes, a Planck mass BH has 1 bit of entropy--i.e., the minimum possible. Another way of putting this is that there is only one possible quantum state corresponding to a Planck mass BH. BH's of larger mass basically just add more bits, meaning more possible states and more entropy. But there's no way to subtract any more bits to get a BH of smaller mass.

However there are some striking similarities between fundamental particles and black holes.

Yes, this whole area is a fascinating field for speculation.

 

[michael879]

a Planck mass BH has 1 bit of entropy

Define 1 bit? (in the sense you are using it)

*edit* also, naked singularities aren't black bodies so they don't obey the same thermodynamic properties as black holes (i.e. they have no temperature)

 

[PeterDonis]

Define 1 bit? (in the sense you are using it)

My terminology here is based on the connection between thermodynamic entropy and information-theoretic (or Shannon) entropy. See here:

http://en.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theory

Basically, the number of bits of entropy is the number of bits of information that would be required to completely specify the system's state, given the macroscopic information you have. For a BH, the "macroscopic information" would be its mass, charge, and spin.

Strictly speaking, a Planck mass BH would have *zero* bits of entropy since it has only 1 possible internal state (and therefore no information is required to specify what state it is in); so I was off by 1 bit in my previous post. (Also, strictly speaking, by "Planck mass BH" I really meant "Planck mass BH with zero charge or spin"--I don't know offhand how the presence of charge or spin changes things at this scale.)

*edit* also, naked singularities aren't black bodies so they don't obey the same thermodynamic properties as black holes (i.e. they have no temperature)

True, the reasoning I gave only applies to BH's, not to naked singularities. So there could still be room for a theory that said that, for example, "fundamental particles" are really Kerr-Newman geometries with charge/spin greater than the "critical value" that produces a naked singularity.

 

[michael879]

Basically, the number of bits of entropy is the number of bits of information that would be required to completely specify the system's state, given the macroscopic information you have. For a BH, the "macroscopic information" would be its mass, charge, and spin.

Strictly speaking, a Planck mass BH would have *zero* bits of entropy since it has only 1 possible internal state (and therefore no information is required to specify what state it is in); so I was off by 1 bit in my previous post. (Also, strictly speaking, by "Planck mass BH" I really meant "Planck mass BH with zero charge or spin"--I don't know offhand how the presence of charge or spin changes things at this scale.)

Why would a Planck mass BH have 1 internal state while a black hole with mass M > M_P would have more? Sorry to be nitpicking, but your logic just seems kind of circular. From the papers I've read I had the impression "1 bit" was defined as the entropy contained in a Planck mass black hole. This number happens to be 4π, so I don't really see why 1 bit would necessary correspond to an entropy of 4π.

True, the reasoning I gave only applies to BH's, not to naked singularities. So there could still be room for a theory that said that, for example, "fundamental particles" are really Kerr-Newman geometries with charge/spin greater than the "critical value" that produces a naked singularity.

Well every fundamental particle found does satisfy M² < a² + Q², so general relativity would describe all of them as naked Kerr-Newman ring singularities. And since we're talking about fundamental particles and black holes, entropy really doesn't come into the equation at all (I've been assuming a naked singularity this whole time).

 

[PeterDonis]

Why would a Planck mass BH have 1 internal state while a black hole with mass M > M_P would have more?

That's one of the $64M questions of quantum gravity. Nobody knows the answer because nobody has a "kinetic" model of BH's that derives their macroscopic properties by statistical averaging over microstates. But we can infer just from the macroscopic facts about BH entropy that the number of internal states must go up with horizon area (and hence with the mass), since the entropy itself does. The fact that entropy = number of bits needed to completely specify the internal state (or, equivalently, the logarithm of the total number of possible internal states) is not specific to BH's; it's a general fact about entropy in any physical system. We just don't know precisely *what* the internal states are for a BH that the macroscopic facts about BH's are derived from.

 

[PeterDonis]

Well every fundamental particle found does satisfy M² < a² + Q², so general relativity would describe all of them as naked Kerr-Newman ring singularities.

Only if one accepts that GR continues to be valid in this regime. Most physicists believe that the singularities are signs that GR, as a theory, cannot cover these regimes; or, equivalently, that GR is only valid up to a certain very high but finite spacetime curvature, above which some other theory has to take over. This is connected to the view of GR as an "effective field theory" that I mentioned before.

 

[PeterDonis]

Well every fundamental particle found does satisfy M² < a² + Q², so general relativity would describe all of them as naked Kerr-Newman ring singularities.

Also, for fundamental particles to actually be Kerr-Newman "super-extremal" BH's, they would have to be confined within a small enough radius--which since their masses are all much, much smaller than the Planck mass, would have to be much, much smaller than the Planck length. But the smallest lengths that we have observed fundamental particles to be confined to are much, much larger than the Planck length. So at best, we could possibly model them as rotating massive objects with r >> M. But it's actually not clear that the exterior vacuum geometry of rotating massive objects is still Kerr-Newman, except asymptotically; there is no analogue to Birkhoff's theorem for rotating spacetimes.

None of this is intended to say that this set of ideas is not very interesting; it is. It's just not as straightforward as it may look at first.

 

[michael879]

That's one of the $64M questions of quantum gravity. Nobody knows the answer because nobody has a "kinetic" model of BH's that derives their macroscopic properties by statistical averaging over microstates. But we can infer just from the macroscopic facts about BH entropy that the number of internal states must go up with horizon area (and hence with the mass), since the entropy itself does. The fact that entropy = number of bits needed to completely specify the internal state (or, equivalently, the logarithm of the total number of possible internal states) is not specific to BH's; it's a general fact about entropy in any physical system. We just don't know precisely *what* the internal states are for a BH that the macroscopic facts about BH's are derived from.

Thats my point, setting the Planck mass as the "1 bit" scale is complete speculation. By saying you can't have a black hole with a mass less than the Planck mass you're making an assumption with no experimental or theoretical evidence. Don't get me wrong, its a very sensible limit, since the Planck scale is where quantum gravity effects should starts to dominate. But to say the minimum mass black hole is exactly 1 Planck mass seems overly restrictive given the limit information we have.

Only if one accepts that GR continues to be valid in this regime. Most physicists believe that the singularities are signs that GR, as a theory, cannot cover these regimes; or, equivalently, that GR is only valid up to a certain very high but finite spacetime curvature, above which some other theory has to take over. This is connected to the view of GR as an "effective field theory" that I mentioned before.

I know all that, all I was saying was that GR would describe fundamental particles as kerr-newman naked singularities. Bringing GR to that scale is highly speculative though, as you mentioned.

Also, for fundamental particles to actually be Kerr-Newman "super-extremal" BH's, they would have to be confined within a small enough radius--which since their masses are all much, much smaller than the Planck mass, would have to be much, much smaller than the Planck length. But the smallest lengths that we have observed fundamental particles to be confined to are much, much larger than the Planck length. So at best, we could possibly model them as rotating massive objects with r >> M. But it's actually not clear that the exterior vacuum geometry of rotating massive objects is still Kerr-Newman, except asymptotically; there is no analogue to Birkhoff's theorem for rotating spacetimes.

Actually you'd be suprised at how big fundamental particles would actually be if described by the Kerr-Newman geometry. The electron, for example, would have a "radius" of a=3x10^-13. Because of the warped spacetime the a parameter isn't really the radius, but 2πa is the circumference, so from far away the radius would appear to be a. The more massive particles would have smaller radii, but not significantly.

Also, I was treating the particles as real ring singularities, not extended objects so the Kerr-Newman geometry would apply.

 

[PeterDonis]

By saying you can't have a black hole with a mass less than the Planck mass you're making an assumption with no experimental or theoretical evidence.

I certainly agree that there is no experimental evidence, but I wouldn't say there is no theoretical evidence. The formula for BH entropy in terms of area in Planck units is not an approximate formula, and it requires a BH with area of 4 in Planck units to have an entropy of 1. I would say that counts as theoretical evidence. (I realize that area of 4 in Planck units does not translate to mass of exactly 1 in Planck units, because there is a factor of pi in there; I didn't actually mean to imply that the "minimum" BH mass on this view was *exactly* the Planck mass, just that that's the right scale.)

Actually you'd be suprised at how big fundamental particles would actually be if described by the Kerr-Newman geometry. The electron, for example, would have a "radius" of a=3x10^-13. Because of the warped spacetime the a parameter isn't really the radius, but 2πa is the circumference, so from far away the radius would appear to be a.

How are you deriving this number? It looks like the electron's Compton wavelength.

 

[michael879]

I certainly agree that there is no experimental evidence, but I wouldn't say there is no theoretical evidence. The formula for BH entropy in terms of area in Planck units is not an approximate formula, and it requires a BH with area of 4 in Planck units to have an entropy of 1. I would say that counts as theoretical evidence. (I realize that area of 4 in Planck units does not translate to mass of exactly 1 in Planck units, because there is a factor of pi in there; I didn't actually mean to imply that the "minimum" BH mass on this view was *exactly* the Planck mass, just that that's the right scale.)

oooo, ok I see what you're saying now. Thats actually an interesting point, since an entropy of O(1) clearly implies O(1) microstates, which will decrease logarithmically. In order to have a microstate of exactly 1, you would need 0 entropy, which doesn't appear to be possible with the entropy formula. Of course non-integer numbers of microstates makes no sense (although it might for a black hole where the microstates are hidden...), so by the time you're in O(1) something must change.

However if non-integer "number of microstates" is possible, the minimum black hole mass is 0 which would correspond to 0 entropy. Whether or not these are possible I don't know. Part of me is saying no, because black holes lose entropy to the outside world through radiation, and in the real world non-integer microstates is not possible. The other part of me is thinking that if you take quantum mechanics into account non-integer microstates may be possible (i.e. continuous rather than discretized entropy). I only know that entropy is discretized classically. I don't really remember quantum statistical mechanics very well.

How are you deriving this number? It looks like the electron's Compton wavelength.

a = J/(Mc), J = h/2π, M = m_e
And yes it is exactly the Compton radius. Just another of the striking "coincidences" I was mentioning earlier

 

[PeterDonis]

a = J/(Mc), J = h/2π, M = m_e
And yes it is exactly the Compton radius. Just another of the striking "coincidences" I was mentioning earlier

Ah, OK, so basically the geometric "radius" associated with the electron's spin turns out to be its Compton radius. Small point: J should be h/4 pi since the electron has spin 1/2.