I Spacecraft With Solar Mass Energy Equivalent Kinetic Energy

[PeterDonis]

Wouldn't it be more proper to perform a Lorentz-boost

There is no such thing as a global Lorentz boost in a curved spacetime.

 

[PeterDonis]

the pure Coulomb field picks up a magnetic component when boosted

Yes, but the reason there are off-diagonal components of the EM field tensor when it is boosted is that there are off-diagonal components of the EM field tensor before it is boosted. The EM field tensor is antisymmetric, so it always has off-diagonal components, even for a "pure Coulomb" field.

The metric tensor in Schwarzschild coordinates, by contrast, is diagonal, and a simple "boost" in one direction will not necessarily add any off-diagonal components. You would have to do the actual math to see.

 

[renormalize]

There is no such thing as a global Lorentz boost in a curved spacetime.

Maybe in a generic curved spacetime, but they do exist in specific spacetimes. As far as I can tell by reading https://arxiv.org/pdf/gr-qc/9805023.pdf, Lorentz transformations can be implemented by first expressing the Schwarzschild (and more generally the Kerr) metric in the Kerr-Schild (KS) form $g_{\mu\nu}\left(x\right)=\eta_{\mu\nu}+2H\left(x\right)l_{\mu}l_{\nu}$ , where $\eta$ is a flat background metric:

and then defining the boost velocity $\mathbf{v}$ relative to that background metric:

The result for the boosted KS Schwarzschild metric turns out to be:

Note that the four non-vanishing components of the null-vector $l$ manifestly give the boosted metric off-diagonal terms, and I'm doubtful that there exists a coordinate transform that returns that metric to a Schwarzschild-like diagonal form. Regardless, the explicit result (23) makes it straightforward, if tedious, to compute the Christoffel symbols and examine the geodesics followed by a test mass in boosted Schwarzschild spacetime. But for now that's above my pay grade. Maybe it's time to break out the computer algebra?

 

[PeterDonis]

Maybe in a generic curved spacetime, but they do exist in specific spacetimes.

No. There is no such thing as a global Lorentz boost in any curved spacetime. There can't be, because the definition of a Lorentz transformation is that it preserves the Minkowski metric on the spacetime--but a curved spacetime's metric is not the Minkowksi metric.

The transformations in the paper you referenced are expanding the usage of the term "Lorentz boost" to apply to coordinate transformations in asymptotically flat spacetimes that, heuristically, "look like" Lorentz boosts at infinity (where spacetime is flat--but infinity is not actually part of the physical spacetime, it's a mathematical artifact used to aid in computations). But that doesn't make them actual Lorentz boosts. It just means the authors of the paper (and other similar papers in the literature) are using terminology in a way that experts have no problem properly interpreting, but that can be misleading if you don't understand what they're actually doing.

 

[PeterDonis]

I'm doubtful that there exists a coordinate transform that returns that metric to a Schwarzschild-like diagonal form

I don't know why you would be. Any valid coordinate transformation is invertible.

 

[renormalize]

No. There is no such thing as a global Lorentz boost in any curved spacetime. There can't be, because the definition of a Lorentz transformation is that it preserves the Minkowski metric on the spacetime--but a curved spacetime's metric is not the Minkowksi metric.

@PeterDonis thank you for the correction; I acknowledge my confusion on this point.

Could you advise me on proper/better terminology? For some specific curved spacetime, coordinatized by a specific collection of four quantities $x$, consider the transformation (as gleaned from the paper cited in post #63): $x\rightarrow x'=\Lambda\,x,\:g\left(x\right)\rightarrow g'\left(x'\right)=\Lambda^{-1}g\left(x\right)\Lambda,\:\Lambda\in\text{SO}\left(1,3\right)$. In lieu of using terms like Lorentz "boost", "rotation", etc., what nomenclature would you suggest to label this type of transformation, i.e., how best to distinguish this operation from the usual sense of Lorentz transforms that preserve the flat metric but don't exist globally in curved spacetime?

 

[PeterDonis]

Could you advise me on proper/better terminology?

Unfortunately I'm not aware of any terminology that's consistent in the literature for particular classes of coordinate transformation in curved spacetimes. One just has to be aware that if terms like "Lorentz boost" are used they can't possibly mean the same transformation that they would mean in flat spacetime, so one has to look at the actual properties of the transformation (such as, in the case we discussed, the fact that it is restricted to asymptotically flat spacetimes and "looks like" an ordinary Lorentz boost at infinity).

 

[Devin-M]

If the shuttle passed a 100kg block of ballistic gelatin at a close approach of 100m, would the gelatin essentially not “feel” the gravitational wave until after close approach, since the gravitational wave and shuttle would be moving at essentially the same speed?

 

[PeterDonis]

If the shuttle passed a 100kg block of ballistic gelatin at a close approach of 100m, would the gelatin essentially not “feel” the gravitational wave until after close approach, since the gravitational wave and shuttle would be moving at essentially the same speed?

Yes.

 

[pervect]

But in these coordinates, while we would certainly expect the particle to be deflected, we would not, it seems to me, expect the system of large masses (the solar system in this case) to be disrupted. We would expect its configuration to be basically the same after the flyby as before--because in this frame the particle has negligible effect on the spacetime geometry.

However, a flyby of either a one solar mass object or a two solar mass object (meaning, the total energy of the object taking into account that in the solar system frame it is moving at a tiny smidgen less than the speed of light) through the solar system would be expected to disrupt the solar system, not just deflect the object.

So we still have a disconnect here: in one frame (the object's rest frame), we expect no disruption of the solar system, but in another frame (the solar system rest frame), we do. So the intuitive reasoning described above must be wrong in at least one frame.

What happens in the spaceship frame did has me puzzled a bit. It appeared at first glance to contradict the paper, but it turns out there is no conflict. I see that Vanadium made similar comments.

I will note that the paper I cited, and my analysis below is for dust, not planets, so I will not address that point that you raised.

However, what happens for the dust case in the frame of the spaceship is interesting, and I think I have a handle on it. The ultra-relativistic dust flies by the spaceship, which has some very small mass, so the dust is deflected only slightly in the frame of the spaceship. Essentially, it should be deflected by twice the Newtonian deflection due to the small, but non-zero, gravity of the spaceship. It's well known that light deflects twice as much in GR as in Newtonian physics, and the difference between the geodesics of light and the geodesics of ultra-relativistic particles is negligible.

Thus we expect in the space-ship frame that the dust will be deflected by some small angle $\theta$, winding up with a velocity in the longitudinal direction of $\beta \cos \theta$ and a transverse velocity of $\beta \sin \theta$, where we can use the usual GR formula to compute $\theta$.

However, to transform from the space-ship frame to a frame where the dust was initially stationary, we essentially need to do a relativistic velocity subtraction.

The velocity subtraction formula with a transverse and longitudinal components is a bit messy, wiki gives it in https://en.wikipedia.org/wiki/Velocity-addition_formula

The relevant formula is that the longitudinal direction is "x" in the wiki analysis, and the transverse direction is "y".

The formula wiki gives is then

$$ u^\prime_y = u_y \frac{\sqrt{1- \beta^2}}{1-\beta \beta \cos \theta}$$

which with $\cos \theta ~ 1$ should basically mean the transverse velocity in the dust frame should be larger by a factor of approximately $\gamma$ than its small value in the spaceship frame.

 

[Devin-M]

Would it be fair to call this a gravitational shockwave analogous to a supersonic aircraft where you don’t hear it coming until after its close approach, and then BOOM (sonic boom) aka shockwave, but in this case a gravitational shockwave?

 

[PeterDonis]

Would it be fair to call this a gravitational shockwave analogous to a supersonic aircraft

No, since there is no such thing as "supersonic" in vacuum--nothing can move faster than light.

There is an electromagnetic analogue to a supersonic shock wave in a medium, where objects can move faster than the speed of light in the medium (but still slower than the speed of light in vacuum). This is called Cerenkov radiation. There might possibly be a gravitational analogue to that in a medium, but in any case, that is not what we're talking about here.

 

[Devin-M]

No, since there is no such thing as "supersonic" in vacuum--nothing can move faster than light.

There is an electromagnetic analogue to a supersonic shock wave in a medium, where objects can move faster than the speed of light in the medium (but still slower than the speed of light in vacuum). This is called Cerenkov radiation. There might possibly be a gravitational analogue to that in a medium, but in any case, that is not what we're talking about here.

So inside a 100kg test block of ballistic gelatin with 100m separation distance at close approach would the analogy be appropriate since in that case there is a medium involved?

 

[PeterDonis]

So inside a 100kg test block of ballistic gelatin with 100m separation distance at close approach would the analogy be appropriate since in that case there is a medium involved?

You said 100m separation distance at close approach, so the shuttle is not passing through a medium, it's passing through vacuum.

 

[Devin-M]

I’m most interested in whether the gelatin stays in 1 piece.

 

[PeterDonis]

I’m most interested in whether the gelatin stays in 1 piece.

Since, as I posted a while back now, gelatin has no structural strength to speak of, I would not expect it to, but that just makes gelatin a bad example. A more interesting question would be whether a block of, say, steel, or carbon nanotubes, would stay in one piece. I think someone would have to run the detailed numbers to answer that.

 

[Devin-M]

A more interesting question would be whether a block of, say, steel, or carbon nanotubes, would stay in one piece. I think someone would have to run the detailed numbers to answer that.

Would it likely be a difficult 3D animation (numerical simulation) to make for someone who already had the appropriate physics software? I’d be interested in seeing the final shape if any for such a steel block.

 

[PeterDonis]

Would it likely be a difficult 3D animation (numerical simulation) to make for someone who already had the appropriate physics software?

The "appropriate physics software" is not something anyone who is not already an expert in numerical simulations in relativity would have. The people who are such experts run their simulations on supercomputers.

 

[Devin-M]

Right now I’m picturing half a pancake.

 

[PeterDonis]

Right now I’m picturing half a pancake.

I'm not sure what you're basing that on, but at this point I don't think we can say much more about the specifics of the result. I don't know that anyone here has access to the relevant numerical simulation capability, and I don't know if there is anything already in the literature that would be helpful for the specific case you're interested in.