Is E=mc^2 a Bidirectional Equation in Physics?

[PeterDonis]

I was thinking that if the photon has a gravitational effect, it would be similar to a matter particle's gravity effect distorting space.

A matter particle's gravity doesn't "distort space". It curves spacetime. "Space" is not a invariant concept. Only spacetime is.

 

[Ibix]

That's a bit strong; there are classical solutions describing how light gravitates and how that affects the motion of test particles.

Buzz Bloom was specifically asking about "a photon" in #49, which is why I answered as I did. I did go on to comment on the classical case.

The pp-wave spacetimes you mention are, IIRC, one of the general classes of null electrovacuum solutions.

...and just for fun I generated the geodesic equations for the Gaussian pulse spacetime mentioned in the ultraboost article I linked earlier. Assuming that I didn't make any silly mistakes, if we assume that the neutron has no motion tangential to the path of the light pulse then:$$\frac{d^2r}{d\tau^2}=-\frac{2amK^2}{\pi r(1+a^2K^2\tau^2)}$$where $K$ is the Killing constant associated with the $v$ symmetry (which is something like the energy minus the momentum of the neutron at infinity, if I'm not mistaken) and $\tau=0$ when it passes the pulse. I don't think there's a closed form solution for $r(\tau)$, unless someone has better differential equation fu than I.

The $r$ coordinate is a radial distance from the axis of the axially symmetric radiation pulse. Its second derivative (i.e., the radial coordinate acceleration) is small for large $\pm\tau$ and at a maximum at $\tau=0$, and is always towards the axis - so the pulse attracts the neutron. It's smaller at large $r$, and smaller for a shorter (large $a$) pulse.

There's also a non-trivial equation for $d^2v/d\tau^2$, but it's even less edifying.

 

[PeterDonis]

the Gaussian pulse spacetime mentioned in the ultraboost article I linked earlier.

I missed that link before. Yes, the ultraboost is a good pedagogical example. The only caveat I would make about it in the context of this discussion is that I'm not sure it's actually a solution of the coupled Einstein-Maxwell equations, so I'm not sure I was correct to say that pp-wave spacetimes are a subset of null electrovacuum solutions.

Another class of solutions of interest here is the null dust solutions, such as Vaidya spacetime, which also are not solutions of the Einstein-Maxwell equations, but are often used to describe gravitational properties of null radiation.

 

[Buzz Bloom]

A matter particle's gravity doesn't "distort space". It curves spacetime.

Suppose the "particle" of interest is a BH in an otherwise empty universe. The time static radial distortion of space around the BH is similar to the concept I have about a "distortion of space". That is, in this case, the ratio of radius to circumference of a circle around the BH varies with radius. I may be mistaken, but I think this is the de Sitter–Schwarzschild metric. Of course my use of vocabulary may also be wrong.

 

[PeterDonis]

Suppose the "particle" of interest is a BH in an otherwise empty universe. The time static radial distortion of space around the BH is similar to the concept I have about a "distortion of space"

And this concept is only applicable to that particular class of spacetimes. The spacetimes which have electromagnetic fields as the source of gravity, as well as the other spacetimes referenced in this thread as possible models for "light as a source of gravity", the ultraboost and null dust spacetimes, are not members of that class of spacetimes, so the concept you describe does not apply to them.

Also, even in the class of spacetimes to which it is applicable, the concept you describe only applies to a particular class of observers, the ones who are "hovering" at a constant altitude above the hole's horizon. Observers who are free-falling into the hole do not see the "distortion of space" you describe.

the ratio of radius to circumference of a circle around the BH varies with radius.

Not quite. The ratio of radial coordinate to circumference is constant: it is $2 \pi$ everywhere, because that's how the Schwarzschild radial coordinate is defined.

For a BH, "radius" in the sense of "physical distance from the center" isn't even well-defined, because there is no "center" in that sense for a BH. For static observers (the ones who are "hovering" at a constant altitude above the horizon), there is a well-defined physical distance from the event horizon (although even that has to be defined as a limiting process, since there are no static observers at the horizon), but it is not the same as the "radius" you are imagining.

I may be mistaken, but I think this is the de Sitter–Schwarzschild metric.

No, it's just the Schwarzschild metric.

 

[Herman Trivilino]

When an electron hits a positron, is it not a possibility that the result is photons?

The electron-positron pair is converted to a pair of photons. The electron-positron pair have energy. The pair of photons have energy.

 

[Ibix]

The time static radial distortion of space around the BH is similar to the concept I have about a "distortion of space".

In an eternal black hole spacetime, it's the time-independence of the metric that let's you define "space" in a non-arbitrary way - you pick the set of slices of spacetime that don't change over time. (That can be stated more formally in terms of Killing vector fields, as I did earlier.)

The problem is that there doesn't appear to be a way to write the metric of the spacetimes we're talking about here that is independent of time, so there isn't a unique notion of unchanging space. Intuitively, this is because the source of gravity is a light pulse. You can't be at rest with respect to a light pulse, so the source of gravity is always moving for everybody, so the gravitational field is always changing for everybody, so there's no notion of unchanging space. That means there's no non-arbitrary way of defining "space", so asking if it's curved or not boils down to whether you chose to define it to be curved or not.

 

[Buzz Bloom]

For a BH, "radius" in the sense of "physical distance from the center" isn't even well-defined, because there is no "center" in that sense for a BH.

How about the distance D between two concentric circles, C₁ and C₂, also concentric with the BH. In flat space without the BH

D = (C₂-C₁)/2π.​

With the BH, D would be greater. I get that the measurement of D might be difficult, but not impossible. One possibility could be based on the time measured for a radial round trip light signal between the circles.

EDIT: Correction of a careless error made to the equation for D. Thank you @Ibix.

 

[Ibix]

In flat space without the BH
D = (C2-C1)/π.

That's fine. I'd suggest that the radial distance between the circles, $(C_2-C_1)/2\pi$ might be a more sensible measure, but that's a minor point.

With the BH, D would be greater.

Pedantically, you need to require that the circles lie outside the event horizon in this case. More seriously, you need to specify how you are defining "space" in order to measure a spatial distance. There is an obvious way to do it in the case of an eternal black hole, as I've mentioned, and you're probably assuming it, but it is an assumption. In that case, the distance is $$\int_{R_1}^{R_2}\frac{dr}{\sqrt{1-\frac{2GM}{c^2r}}}$$where $R_1=C_1/2\pi$ and similarly $R_2$.

I get that the measurement of D might be difficult, but not impossible. One possibility could be based on the time measured for a radial round trip light signal between the circles.

You can certainly use round trip light times, but you aren't required to interpret the travel times in terms of spatial curvature. You can interpret it in terms of varying speed of light if you like.

 

[PeterDonis]

How about the distance D between two concentric circles, C₁ and C₂, also concentric with the BH. In flat space without the BH

D = (C₂-C₁)/π.​

With the BH, D would be greater.

Yes, that's correct, as long as both circles are outside the horizon.

 

[PeterDonis]

you need to specify how you are defining "space" in order to measure a spatial distance.

Yes, and the underlying assumption being made in this case is that the "distance" being measured is the distance that would be measured by stationary rulers, i.e., rulers that are all "hovering" at a fixed altitude above the black hole. Or, if we want to be really pedantic and eliminate the effects of proper acceleration on the rulers, we can arrange to have a family of free-falling rulers all come to rest momentarily relative to the two circles, in just the right way to lay end to end along the radial line between them.