I Spacecraft With Solar Mass Energy Equivalent Kinetic Energy

[Ibix]

I don't see how they could be since they are explicitly giving a nonzero active gravitational mass to the object

Are they? I think pervect's description of the paper is that they fire an array of test particles at a Schwarzschild metric with mass $M$ with "speed at infinity" $\beta$ and look at the angles between the asymptotes of the orbit. The particle mass doesn't appear anywhere in the abstract, anyway. So I think only one of their objects is gravitationally active; here we're at least considering the possibility that both are.

 

[Vanadium 50]

Since we have decided this is A level....

It's not that the factor of 2 might be "close enough". It is conceptually wrong. It is implicitly saying that the g₀₀ part of the metric must be considered but not the equally large g₁₁ part. If it is saying anything at all, it's saying we only need to worry about spacetime curvature in the time direction, not in the spatial directions.

(If it is saying anything relativistically at all, of course. It looks to me more quasi-Newtonian - i.e. "GR is the same as Newtonian gravity, provided you plug in a different number for mass")

 

[PeterDonis]

Are they?

The abstract, which is all we have, gives an explicit expression for what they call the active gravitational mass of the object. That expression is not zero.

The particle mass doesn't appear anywhere in the abstract

Yes, it does. The formula explicitly given in the abstract is $\gamma M \left( 1 + \beta^2 \right)$.

 

[PeterDonis]

Since we have decided this is A level

I've split the difference with the OP and changed the thread level to "I".

 

[PeterDonis]

I think pervect's description of the paper is that they fire an array of test particles at a Schwarzschild metric with mass $M$ with "speed at infinity" $\beta$ and look at the angles between the asymptotes of the orbit.

But if we are going to translate that to this scenario, we would have to combine this solution with the metric of the solar system. And the question is how much doing that would change the final metric from the one we already have for the solar system. Our intuitive answer in this thread has been "not very much". But if we take the formula in Olson & Guarino's abstract at face value, as I said before, that would not be correct.

The problem I have with that face value answer, however, is that if we switch to the shuttle's rest frame, it is obvious, as you pointed out, that the shuttle's effect on the overall spacetime geometry is miniscule. And that can't change just because we changed frames--this is just another version of the answer (given in a number of previous PF threads) to the common question of why an object doesn't turn into a black hole if it goes fast enough.

Furthermore, again if we look at things in the shuttle's rest frame, what effect would we expect on the shuttle due to the solar system? Say due to the Sun, to make things simpler. Olson & Guarino's formula, taken at face value, says we should expect the Sun's effect to be $\gamma M_S \left( 1 + \beta^2 \right)$--in other words, a huge effect compared to the Sun's effect when at rest. At the gamma factor we are talking about, the result of that formula is something like $10^{20}$ solar masses, i.e., comparable to the total mass in our observable universe. Is that really the correct answer? It can't be that simple, because all that mass concentrated into a volume like that of the Sun would indeed be a black hole, but we already know that is not the correct answer.

As I say, I haven't looked at the detailed math, either in the Olson & Guarino paper or on my own, so all this is just intuitive. But it seems to me that there must be a disconnect somewhere since what the Olson & Guarino paper appears to say is so different from our intuitive answer in this thread.

One possible resolution of the disconnect might come from asking, where did all the energy come from to boost the shuttle to such a huge gamma factor? Suppose, for example, that we detonated a 2 solar mass star in such a way that it boosted two shuttles, each to the same gamma factor, in opposite directions (so total momentum in the original star's rest frame remains zero). We would have expected the original star to have a significant effect on the spacetime geometry around it, so it seems like we should also expect the two shuttles thrown off in the explosion to have a significant effect on the spacetime geometry surrounding them.

Then the real issue might be the specific form of the effect. The fact that the shuttles are each moving at a tiny smidgen less than the speed of light, in the original star's rest frame (which means, to a good approximation, in the rest frame of any other star systems they pass through) means that they are only within a short enough distance to have a significant effect for a short time--short enough that the overall effect remains small, as our intuitive answer in this thread has it. (That is one way of heuristically describing the effect of the momentum components of the stress-energy tensor that I talked about earlier.) In other words, instead of being a nice spherically symmetric Schwarzschild field as the original star's field was, we now have two narrow "world tubes" filled with stress-energy whose effects will be very different from those of the original star.

 

[Devin-M]

One possible resolution of the disconnect might come from asking, where did all the energy come from to boost the shuttle to such a huge gamma factor?

So if you start with 1 star size clump of matter and one star sized clump of antimatter, and somehow convert almost all of the annihilation energy into 2 shuttles going in opposite directions, what happens to the gravity of the 2 star sized masses which “disappeared?”

 

[Devin-M]

We would have expected the original star to have a significant effect on the spacetime geometry around it, so it seems like we should also expect the two shuttles thrown off in the explosion to have a significant effect on the spacetime geometry surrounding them.

If it was approaching an everyday object could this lead to spaghettification from tidal forces? Imagine it passes a 100kg block of gelatin in perfect vacuum at a close approach of 100m. Is that gelatin staying in 1 piece?

 

[PeterDonis]

So if you start with 1 star size clump of matter and one star sized clump of antimatter, and somehow convert almost all of the annihilation energy into 2 shuttles going in opposite directions, what happens to the gravity of the 2 star sized masses which “disappeared?”

The stress-energy is now all in the shuttles, so that's where the "gravity" is. But, as I said, it will look very different from the gravity of a 2 solar mass star.

 

[PeterDonis]

If it was approaching an everyday object could this lead to spaghettification from tidal forces?

No. We are not talking about a black hole.

Imagine it passes a 100kg block of gelatin in perfect vacuum at a close approach of 100m. Is that gelatin staying in 1 piece?

Gelatin is a bad example because it has no structural strength to speak of, so even a very mild perturbation might cause it to fragment.

I would expect the general effect on objects that do have a reasonable structural strength to be a more or less impulsive (i.e., over a very short time) change in velocity as the shuttle passes. Since the shuttle is moving at only a tiny smidgen less than the speed of light, it will pass any ordinary object very quickly, so there will be no time for effects to build up.

 

[Vanadium 50]

very quickly

Of order an hour.

It will have comparable force to the sun at about 11 light-minutes coming in and 11 light minutes going out, making 22: 23 with rounding. You get half the solar effect at √2 farther out, and a quarter of the effect at a factor of 2, and so ojn. So the scale is around an hour.

We're talking 1/10,000 of Earth's orbital period, so we would expect 0.01% changes to the orbit. That's 10,000 miles. I have no idesa how to calculate what would happen if this happened, but I suspect it would wreck everybody's day.

 

[PeterDonis]

Of order an hour.

For passing through the inner solar system, yes. But for the scenario described in post #37, involving an ordinary object, it will pass in a fraction of a second.

 

[PeterDonis]

It will have comparable force to the sun

Will it? That's the question. It has a solar mass of kinetic energy, but it also has a solar mass of momentum, which should cancel at least a lot of the effect. At least, that was the intuitive guess I gave earlier in the thread. I realize the Olson & Guarino paper appears to be saying something different, but we haven't seen the details of the argument, and what they describe in their abstract is not a long distance effect like the Sun's field, it's an impulsive effect on test particles in or close to the path of the high speed object.

 

[Devin-M]

So comparing to an astronaut’s infall to a solar mass black hole, wouldn’t they expect to be spaghettified and moving close to speed of light when 100m away from the black hole’s event horizon?

Is there an intuitive answer why the tidal forces experienced by an astronaut should be less when the shuttle passes 100m away than the black hole case and not result in spaghettification?

 

[Vanadium 50]

but it also has a solar mass of momentum, which should cancel at least a lot of the effect.

It doesn't cancel it. It doubles it.

Olson & Guari do the full calculation, but you already know the answer. g₀₀ and g₁₁ are nearly equal and the only non-zero terms in the metric. What else has a metric like that? Light.

Just as light have a factor of 2 more bend than if you just consider g₀₀, so must the relativistic shuttle - the same equations have the same solution. And since momentum is conserved, you have twice the recoil on the earth.

You'd expect this approximation to be good to of order (m/E), and 2 is close to Olson & Guar's 1+β².

The question of what happens to the Earth only makes sense if this happens far away. If you drag two solar masses worth of "fuel" anywhere near the earth, of course the orbit will be highly perturbed. Further, if the "anti-shuttle" is anywhere near the shuttle, the total g₁₁ gets small and we are back to the Schwartzchild metric.

 

[Vanadium 50]

why the tidal forces experienced by an astronaut should be less than a solar mass black hole case at the same 100m distance?

Why do you think a fast moving object should behave the same as a slow heavy object. Does a bullet act like a boulder?

 

[PeterDonis]

comparing to an astronaut’s infall to a solar mass black hole

Which makes no sense because the two configurations are so different.

 

[PeterDonis]

It doesn't cancel it. It doubles it.

It doubles it in the narrow world tube surrounding the path of the shuttle, yes. That corresponds to doubling the bending of the path of a fast moving test particle by the Sun as compared to the Newtonian value, because the test particle passes very close to the narrow world tube of the Sun.

But does the field outside that world tube still fall off as the inverse square of the distance with an effective gravitational mass of $\gamma M \left( 1 + \beta^2 \right)$? That's the part I'm wondering about. It seems to me that it should fall off faster.

 

[Vanadium 50]

field outside that world tube

I don't know. I don't see an obvious limiting case that can be solved by symmetry and not the full solution. I suspect one or both bets are dotted into the line of sight direction at large distances.

Insofar as Ives-Stillwell can be used as an analogy, at large transverse distances you get a 1, not a 2.

 

[pervect]

The abstract of that paper, though, shows a factor of $\gamma$ in what they call the "active gravitational mass", i.e., taken at face value, they seem to be saying that the relativistic mass is the source of gravit, with an extra GR factor of $1 + \beta^2$ that comes in for similar reasons as in the analysis of light bending by the Sun.

However, they are defining "active gravitational mass" in a very narrow way: it is "measured by the transverse (and longitudinal) velocities which such a moving mass induces in test particles initially at rest near its path". That is not the same as the kind of "gravitational effects" that the OP is asking about.

I have so far been unable to find a non-paywalled version of the Olson & Guarino paper (for example, a preprint does not appear to be on arxiv.org), so I can't comment on the details of their reasoning. But I would be cautious about interpreting it with regard to this discussion.

I've seen a version of the full Olson & Guarino paper, but I'm not sure it's available anymore. So my comments are going mostly by memory.

However, I think the abstract is sufficient to answer the OP's question, and the answer is that the effect of the flyby (such as orbital pertubations) of such an ultra-relativistic object wil be simliar to a Newtonian flyby of an object with twice the mass. The factor is 2 because the flyby velocity beta is essentially unity.

The actual paper basically calculates the Newtonian and GR effects of a flyby on a cloud of dust, looking at the total induced velocity in the flat spacetime after the flyby.

While the actual flyby will cause additional GR effects such as the emission of gravitational waves that aren't present in the "dust" model, I don't think they'll be very significant. The end result of the flyby will be similar to a Newtonian flyby of not a 1x solar mass, but 2x a solar mass.

Interestingly enough, it's not a trivial calculation to find the effects of a Newtonian flyby. But I'll leave that to the reader.

Many people on PF don't like the fact that the paper's authors uses "relativistic mass" rather than "energy". But it's just a different name for the same thing, it doesn't affect the results. Since the paper compares the GR results to Newtonian results, and Newtonian physics uses mass, the author's choice to use "relativistic mass" rather than energy is understandable.

A non-rigorous popular level "explanation" of the factor of 2 is that light passing by a massive object deflects twice as much in GR as it does in Newtonian physics. The deflection of an ultra-relativistic particle moving at almost the speed of light is similar. The effect is essentially viewing this scenario from a set of coordinates where the ultra-relativistic particle is nearly stationary, and the large mass is moving instead.

 

[pervect]

While I don't recall how the paper did the calculation anymore, what comes to mind is considering the Schwarzschild solution in isotropic coordinate (t,x,y,z), perform a coordinatge transformation to cylindrical coordinates, (t, r, theta, z), then perform an additional coordinate transformation of z' = z - $\beta$ t to make the large mass "move" in the z direction in the transformed coordinates.

While one could also use isotorpic Schwarzschild (t,x,y,z) coordiantes directly, and do the z' = z-$\beta$ t coordiante transformation without first going to cylindrical, the cylindrical coordinates will be simpler as the problem has cylindrical symmetry. The basic insight is as my previous response suggested, it's just a messy coordinate change of the problem of the deflection of an ultra-relativistic particle. Then the particle follows a geodesic in the transformed coordinates.

It's likely that one would only want to consider the linearized problem - so rather than using the full non-linear isotropic solution, one just considers the linearized version thereof.

 

[PeterDonis]

he effect is essentially viewing this scenario from a set of coordinates where the ultra-relativistic particle is nearly stationary, and the large mass is moving instead.

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.

 

[Vanadium 50]

We would expect its configuration to be basically the same after the flyby as before--

Remember, the solar system is Lorentz contracted so that its width is smaller than a proton. An ultra-microscopic change in this frame ("basically the same") can correspond to a large change in the original frame.

In this case, planets move by of order 10,000 miles.

 

[PeterDonis]

the solar system is Lorentz contracted so that its width is smaller than a proton

It's contracted along the direction of motion, but not transversely. And it is flying by the shuttle in this frame at just a tiny smidgen less than the speed of light. So I'm not sure why there would even be an ultra-microscopic change to the solar system in this frame. It's certainly not obvious to me why planets would be expected to move transversely by 10,000 miles.

Of course, all these are intuitive arguments, not math. If I can't find an accessible copy of the Olson & Guarino paper online I will have to try to work through some math myself if I get a chance.

 

[Vanadium 50]

t's certainly not obvious to me why planets would be expected to move transversely by 10,000 miles.

So you have an argument that the initial motion is longitudinal. Not that it is small.

This is SR, not GR. You have the same issue in electron-proton scattering. In the electron's frame the proton is a static pancake. Nonetheless, quarks can still scatter.

 

[PeterDonis]

So you have an argument that the initial motion is longitudinal.

No, it's a reason for doubting that objects at significant transverse separations will have large motions induced. Obviously if the shuttle impacts the Sun or a planet head on we will have a significant change. But we're talking about a flyby, where the shuttle doesn't come close to the Sun or any of the planets. At least, that's how I'm interpreting the OP's question of whether one solar mass of kinetic energy is significant--he's asking whether it's significant at ordinary planetary distances, the way the Sun itself is.

This is SR, not GR.

No, it isn't. The whole point is that we have to consider the shuttle as being a non-negligible source of gravity, not just the solar system. Otherwise the answer is obvious: a test particle by definition cannot induce motion in other objects, no matter how high a gamma factor we boost it to.

You have the same issue in electron-proton scattering.

No, it's not at all the same. For electrons to scatter quarks inside protons, the electrons have to be shot inside the protons. AFAIK electrons passing by protons at significant transverse distances do not scatter quarks inside the protons, even if they have huge gamma factors.

 

[Vanadium 50]

I can tell you are getting angry, and I know where that leads. However, before I go, this has all been worked out in the 30's. The words to look up are "Breit frame". You can see how this works out in deeply inelastic scattering by hunting down a copy of George Sterman's lecture notes from various CTEQ summer schools. He's at Stony Brook, so that might help Google find them.

 

[PeterDonis]

The words to look up are "Breit frame".

I'll look it up when I get a chance.

 

[PeterDonis]

I can tell you are getting angry

No, just not convinced--or more precisely, not sure at this point which of the two intuitive arguments I've described, that give different answers, are giving the right answer. But I've already said that I need to take the time to work through the math when I get a chance. I think I have enough pointers at this point to do that when I'm able.

 

[PeterDonis]

I know where that leads

I'm not considering demodulating anybody, if that's what you mean.

 

[renormalize]

While I don't recall how the paper did the calculation anymore, what comes to mind is considering the Schwarzschild solution in isotropic coordinate (t,x,y,z), perform a coordinatge transformation to cylindrical coordinates, (t, r, theta, z), then perform an additional coordinate transformation of z' = z - $\beta$ t to make the large mass "move" in the z direction in the transformed coordinates.

It's likely that one would only want to consider the linearized problem - so rather than using the full non-linear isotropic solution, one just considers the linearized version thereof.

I've been thinking along similar lines. But I question your suggested coordinate transformation. Wouldn't it be more proper to perform a Lorentz-boost to get the mass moving in the $z$-direction, à la what Jackson does in Classical Electrodynamics (chap. 11) to get the EM field of a uniformly moving point charge? And just as the pure Coulomb field picks up a magnetic component when boosted (due to mixing of the $(t,z)$ coordinates), I believe a boost of the diagonal Schwarzschild metric field must similarly lead to non-removable off-diagonal terms. So the calculation and interpretation of the boosted metric and its geodesics is apt to be non-trivial. But I do agree that using the linearized Schwarzschild solution may be easier and should suffice if we concentrate on the behavior of geodesics in the "far-field", i.e., those with a large-enough impact parameter $b$.