Kerr, Reissner–Nordström, and Kerr-Newman

[WrathfulSpatula]

We've had Schwarzschild metric in this fork, for a long time. This is a static, stationary black hole. We should expand the treatment to also encompass rotating and charged black holes.

Hypothetical "new physics" proposed by Strano can be turned off entirely, for Schwarzschild. The same will be true for rotating and charged black hole metrics.

It's been a while since I formally considered how my own work (Strano's) might apply to spinning and charged black holes, but the evaporation of a (spinning) Kerr metric ergosphere happens via the ostensible spin-2 graviton, not a quasi-scalar mode from allowing metric tensor torsion, if I recall. Charged Reissner–Nordström black holes are assumed by the author to emit charge in quasi-scalar packets that are sub-"extremal," and these probably happen to carry multiples of 2 fundamental units of charge in a packet. (This includes the possibility for net 0 charge, in which case they very quickly self-annihilate. The author thinks magnetic binding forces are sufficient to keep these other theoretically "bare" like-sign charges coincident, but, rather, they therefore must carry sufficient energy to avoid being "hyperextremal." in order to exist as "real" particles.)

I'll be working on this, at night over the next few weeks, or however long it takes.

[WrathfulSpatula]

To clarify, in (spinning) Kerr metric, the quasi-scalar emission from the interior event horizon undergoes a Penrose process in the ergosphere, separating the quasi-scalar to emit net fundamental spin-2 gravitons. We'll handle this case first, starting tonight.

[WrathfulSpatula]

We have an approximation to the Kerr metric, from #51: we treat Kerr as a Schwarzschild space-time whose preferred frame of accelerated rest locally rotates with omega parameter calculated as in Wikipedia:

https://en.wikipedia.org/wiki/Kerr_metric#Frame_dragging

According to the linked section of the article, Kerr is strictly diffeomorphic to Schwarzschild under the local parameterization by this omega.

First, we approximated omega by the starting point of a finite difference time step, in the MonoBehavior.Update() loop. Tonight, we improved the approximation by averaging the starting and ending omega value, before and after Schwarzschild metric co-motion for the same time interval. (Better approximations, elliptical or hyperbolic, are possible via spline or integration over the time step.) RelativisticObject instances and the player also intrinsically spin, in co-motion with omega.

The Kerr optics still aren't quite right. We might modify our Schwarzschild strong-lensing-at-a-distance blitting shader with a relativistic Doppler shift for tangential velocity from omega. For the full-screen blitting shader, which assumes that the black hole is very distant, the effect would be calculated on a plane orthogonal to line-of-sight from player to "world coordinates" origin, but also depending on the angle of this plane with the axis of Kerr spin.

[WrathfulSpatula]

More on Kerr lensing: Much has been written about the Kerr lens equations, such as this article and many others, though this topic seems like it might still be a research-level problem, rather than neatly wrapped up in a canonical analytical form.

Not having had time peruse the literature, yet, the article linked above starts with the simpler cases of looking directly at a Kerr black hole on the poles, then equatorial plane, before attempting total generality. We can borrow these same cases, under the assumption that Kerr metric is diffeomorphic to Schwarzschild under a map by the local omega angular velocity parameter, of its "frame dragging."

Intuitively, under this assumption, looking directly along the axis of rotation of Kerr metric, one expects a Schwarzschild lensing effect diffeomorphically transformed by a "corkscrew" effect from the integration of omega over the path between source and observer, under these perfectly aligned conditions. The focal point does not change. Any energy difference in the photons is cyclically cancelled as with Schwarzschild metric, between a distant source and a distant observer.

First, I will handle the case of integrating this omega parameter as a diffeomorphism on Schwarzschild, causing this "corkscrew" effect. Then, I will consider the equatorial plane case, including Doppler shift. Then, I'll try to put together an approximation for rotating the observer's viewpoint between these two extremes. (The third spatial dimension to consider is only the rotation of the on-face plane, for the old Schwarzschild lens shader, which is comparatively "trivial.")

[WrathfulSpatula]

In the case of the equatorial plane, approximating by the locally varying omega parameter on Schwarzschild, the effect is probably to shift the source point of the focus toward the away-spinning side (relative the observer) of the black hole, dependent upon black hole spin and inclination angle. Additionally, there might be Doppler blue-shift on the toward-spinning side, (toward the observer,) and red-shift on the away-spinning side.

We'll form the general approximation basically by spherically interpolating between these on-pole and on-equator cases, while, again, the third spatial dimension is just (Euclidean) rotation of the on-face blit plane of the shader.

[WrathfulSpatula]

We've implemented this, as described above. Let's have a word of recap.

Gravitational lensing by Schwarzschild is commonly simplified in approximation to the assumption of a point-like deflector in front of a very distant lensed background, with the observer at a specified finite distance.

If the angle of deflection is 4GM/(rc^2), then this is the same as -4/c^2 times the gravitational potential. Thinking of the deflection as due to interaction with a conservative gravitational mass field potential, we expect equivalent interaction with a conservative gravitational angular momentum potential, directly proportional to fraction of angular momentum over Schwarzschild radius, in Planck units.

Classical intuition tells us this angular momentum potential interaction is entirely and exactly polarizing, (i.e., "corkscrewing,") looking directly along the Kerr axis of rotation; the equatorial term implies the same conservative angular momentum potential interaction magnitude, but orthogonally non-polarizing, for deflections exactly planar to the equator, dropping off as cos(inclination).

Hopefully, it's a reasonable theoretical approximation.

[WrathfulSpatula]

I've been wrestling since last night with whether the maximum deflection in the equatorial plane is exactly the same as along the axis, or whether different by some factor like 2 or 4. I think I have a better argument, that these two are the same.

Consider 2 path integrals of omega approximating "straight lines" in the coordinates, one along the axis of Kerr spin, and the other in the equatorial plane. The maxima of the integrand functions should exactly match for these two, since they are effectively the same point, for \omega. The scaling factor between the two integrated areas would be the scaling between spin-axis-orientation and equatorial-orientation.

Along the spin axis path, the value of \omega follows a half period of cos(inclination). If the integrand takes a maximum value of 1, then the area of the integral is 2. \omega is always entirely at a right angle to the "velocity" of a path along the spin axis.

Integrating a path straight through the (assumed point-like) deflector, in the equatorial plane, the path integral's "velocity" is no longer at a right angle to \omega; instead, it sinusoidally varies, as with the spin axis path. The fixed \omega dragging along the equatorial plane has sin(azimuth) and cos(azimuth) components, parallel and perpendicular to the test path. Integrating both these, the contribution from sin(azimuth) cancels over -Pi/2 to Pi/2, with cos(azimuth) surviving. Also, at the common maximum of the integrands along spin axis and in equator, cos(0)=1, (with the other component being sin(0)=0,) setting the scale for areas.

By heurism, holding the amplitude of the common point between the two integrands fixed, cos(azimuth) and cos(inclination) have the same definite area over the same domain. Hence, I think the integrated areas scale is 1:1, as is already implemented in the shader.

[WrathfulSpatula]

Note that there is no new Doppler shift effect for Kerr, beyond Schwarzschild, since this is a function of the energy change of photons being lensed, which is strictly a function of the conservative mass potential field, not the conservative angular momentum potential field.

[WrathfulSpatula]

By the way, we find a happy, self-consistent result, for a conservative angular momentum potential field, when we consider arbitrary rotation of the spin axis: no matter what angles the observer's position forms with the spin axis and equator, the magnitude of deflection by the the gravitational effect of angular momentum is exactly the same, and depends only on the distance of the observer from the center of the metric. However, the partitioning into orthogonal polarizing and non-polarizing terms depends upon inclination angle.

[WrathfulSpatula]

With Kerr metric sufficiently handled for now, I'm thinking about how to address Reissner-Nordström, for a charged black hole.

Sorry for the 20-year-old Geocities-type page link, (from University of Colorado, Boulder,) but, then, Reissner-Nordström has been known for nearly the full 100+ years since the inception of the theory of general relativity:

https://jila.colorado.edu/~ajsh/bh/rn.html

What's made Kerr metric so easy to handle has been its diffeomorphic mapping to Schwarzschild metric via the locally varying \omega parameter, for angular velocity of frame-dragging. With an implementation of the Schwarzschild ConformalMap type in hand, and the \omega parameter, we can treat Kerr metric as just a very terse transformation layer basically laid on top of Schwarzschild, (or endpoint-averaged over a finite difference time step, as has ultimately become our approach). To make any ConformalMap work, we need free fall coordinates for ComoveOptical(), (and a map between "game world" and free fall coordinates,) and it'd be great if we could directly use Schwarzschild as a base layer for this comovement transformation, again, as with Kerr.

The link above points out that Reissner-Nordström free fall coordinates take the same form as Schwarzschild, though, if we think of the black hole's mass as locally varying over the radial coordinate. (M(r) = M + Q^2/(2*r))

Then, if we modify our Schwarzschild object to require a "position-in-world," "piw", in order to determine the Schwarzschild radius, then we can just treat the free fall coordinates virtually as if they were Schwarzschild!

I might have a working implementation tonight, on this basis.

[WrathfulSpatula]

With the last couple of commits, Reissner-Nordström turns out to be even much easier than Kerr.

Every RelativisticObject acts locally as if Reissner-Nordström were exactly Schwarschild, with an effective Schwarzschild radius parameterized on the radial coordinate. (For a finite difference time step, we lose some comovement accuracy due to the effective radius changing continuously over the course of the time step, though we could take multiple partial samples and compose the comovement over the proper time interval to alleviate this, but we don't necessarily need to for "video game accuracy" purposes.) This includes the player perspective effective radius, which determines how the Schwarzschild radius looks for the shader.

[WrathfulSpatula]

Our speculative black hole evaporation, in this case, implies that the gravitational field can carry charge. (Might as well call these particles "gravitrons.") It has to, for this sort of evaporation to work, and this is where we get really far out, from accepted canonical theory, admittedly.

By Fermi's golden rule, due to the implied extra particle species' Feynman diagrams, we have an additional (overall charge-neutral) evaporation term, where pairs of "gravitrons" carrying net neutral charge immediately self-annihilate to produce a thermal gas of photons, a true luminous "glow." This "should" technically be there in Schwarzschild and Kerr, as well, (if we follow the author's hypothesis,) but we'll reserve it for Reissner-Nordström and Kerr-Newman, (both being charged black hole metrics). At least, this is a reasonable "separation of concerns," from a software development perspective. As always, turning off black hole evaporation will turn off the effect, when we implement it.

(The only other note I'll offer, on this particular theoretical point, is that this additional "gravitron" field mode term, evaporating into photons, should be visible in the thermal spectrum of all stellar black hole candidates we know, whereas active galactic nuclei are too large to exhibit it with any significant intensity. Of all the candidates we have observed, for black holes formed from stellar collapse, these candidates tend to be in binary systems, with a partner, and we have microwave spectra observation data for most of them. While the preferred theory for "microwave excess" in these spectra is synchrotron radiation, if there even is an "excess," per se, Wien's displacement law on the back of a literal envelop doesn't give a result entirely beyond the realm of plausibility, or even at all, for my favorite pet competing explanation.)

[WrathfulSpatula]

We're going to combine the mass, spin, and charge pieces we already have into Kerr-Newman metric first, though, and worry about that later in the weekend. The previous comment is a matter of personal satisfaction with the already non-canonical doEvaporate checkbox behavior.

[WrathfulSpatula]

That's done, but I realize that I missed the effective displacement of the Schwarzschild-like (inner) event horizon for Kerr. Luckily, this is easily handled like Reissner-Nordström.

[WrathfulSpatula]

I misread this section of the Wikipedia article on Kerr metric:

https://en.wikipedia.org/wiki/Kerr_metric#Frame_dragging

The diffeomorphic relationship to Schwarzschild is directly via components of the metric tensor, which I have not used as the effective Schwarzschild radius. While replacing the effective radius with the appropriate term from these coordinates will serve for comovement, it might not match the expectation for optics, as the gravity lens shader calculates.

I will debug.

[WrathfulSpatula]

Sorry for the commit flailing, but there is incremental progress. Kerr is much more difficult to analyze, compared to Reissner-Nordström.

For Reissner-Nordström metric, it's nearly "trivial" to treat it as Schwarzschild metric with a locally varying Schwarzschild radius parameterized on the radial coordinate. Optically, the player sees the Schwarzschild radius parameterization value for the player's own immediate "world (3-)position." At any point in the metric, on a given time coordinate hypersurface, we use the appropriate Schwarzschild radius parameterization value for the RelativisticObject "world 3-position."

We can't just immediately extend this kind of parameterization "trick" to Kerr, maybe at all. Also, of all the coordinates we could define on the Kerr metric, even if the mechanical paths are diffeomorphic to the correct geodesics, we might not have the right coordinates to show the player how the mechanics would "really" look, from the player's perspective.

But, unflagging, we go ahead and try to adapt the parameterized effective Schwarzschild radius approach from Reissner-Nordström, but instead for Kerr. Arbitrarily, say that we rescale just by the respective time coordinate differential factors, between Kerr and Schwarzschild: 1-(r_s)*r/sigma = 1-(r_s')/r. sigma depends on the inclination relative the Kerr equator; this parameterization of an effective Schwarzschild radius is equal to the invariant mass Schwarzschild radius directly along the axis of rotation. So far, it's plausible that, at exactly this axis of perfect symmetry, an effective Schwarzschild radius appears the same as the invariant mass Schwarzschild radius, since the frame dragging velocity approaches 0. Further, our effective radius parameterization responds to the inclination.

Straight from the Kerr metric Wikipedia article, we have an expression for the scale of the time-like coordinate for an observer at infinity relative the first-person proper time of an object closer to the spinning black hole:

This is analogous to the familiar t'/tau=1/(1-r_s/r)^(1/2) for Schwarzschild metric, which we've used (and abused) to approximate the behavior of Kerr up to the omega angular velocity Killing horizon. If we put one expression over the other, to form t/t', we get a coordinate scaling factor between time-like coordinates for an observer at infinity for Kerr, over that for an observer at infinity for Schwarzschild. Note that this time scale is not generally 1:1 at the Kerr poles, even though our effective Schwarzschild radius is! This makes a certain amount of sense, though, as similarly to how our discussion of conservative mass potential and angular momentum potential fields for the optical lensing of Kerr went, earlier, our "mass potential" effect is augmented by a nonzero "angular momentum potential" whose magnitude is independent of observer angle, we found for lensing; there's intrinsic magnitude of angular momentum gravity depending only on radius.

So, we endpoint-average the Killing horizon angular momentum on top of this to finish our comovement and local effective acceleration calculations, right? I guess, but I need better tests. There's a certain amount we can get away with, in numerical finite difference simulation, as opposed to the symbolic math, but I need some distance from this to form a plan for some kind of acceptable validation, even if just via symbolic derivation.

[WrathfulSpatula]

Up to better validation, all of these ConformalMap types are done. I'm closing this issue, while I consider how to address validation in general.