Speed of light weirdness

[peadar1987]

Hey, some questions about general relativity and such:

-So objects moving faster appear to have more mass. Does that mean that, in theory, if you had infinite energy, and accelerated a small spacecraft up to 99.99999% of the speed of light, it would appear to have tremendous mass to any objects it passed by, perturbing their orbits? I'm guessing you wouldn't be able to have a moon orbiting a fast moving person or anything like that, because reference frames, but what would the effect be?

-On reference frames, I'm not 100% clear on how they work. If I am on a "carrier" spaceship in earth orbit capable of 60% of the speed of light, and I hop onto a smaller spaceship and accelerate away, I won't be able to reach the speed of light relative to the original ship, because that would take an infinite amount of energy. However, what if I was to accelerate away to 60% of the speed of light, and the original ship was to do the same thing in the opposite direction? In the reference frame of the earth, neither of us has broken the speed of light, but taking either me or the carrier as your reference frame, we have. Is this possible? How would it work in practice? (I assume that even if this exact case is impossible, similar things happen for subatomic particles, for example, in the LHC, protons at opposite sides of the accelerator ring will have relative velocities higher than the speed of light)

-Gravity can be described as a warping of spacetime by massive objects, resulting in a force. Do other forces work this way? Can the electromagnetic force be described as a warping of some all-pervasive field, or is it sufficient to describe it more "conventionally"?

Thanks for the responses, I'll try my hardest to get my head around them!

[K^2]

-So objects moving faster appear to have more mass. Does that mean that, in theory, if you had infinite energy, and accelerated a small spacecraft up to 99.99999% of the speed of light, it would appear to have tremendous mass to any objects it passed by, perturbing their orbits? I'm guessing you wouldn't be able to have a moon orbiting a fast moving person or anything like that, because reference frames, but what would the effect be?

Not really. Gravity isn't just about mass. A fast-moving object has high energy, but it also has a very high pressure term in the stress-energy tensor. So the net effect is insignificant. A simple way to think of it is that if you have a very, very fast baseball fly past you, from perspective of the baseball it's you who is moving near speed of light. Would your trajectory be affected by a baseball if you're flying at near speed of light by it? No. It's just a baseball. So neither would a very fast baseball have any effect on you.

-On reference frames, I'm not 100% clear on how they work.

Read this

-Gravity can be described as a warping of spacetime by massive objects, resulting in a force. Do other forces work this way? Can the electromagnetic force be described as a warping of some all-pervasive field, or is it sufficient to describe it more "conventionally"?

Both. If you just want to describe electromagnetic field, Maxwell's Equations are sufficient. Unlike Newtonian gravity equations, Maxwell's Equations are exact. However, if you want to describe interactions between electromagnetic field and matter, you might have to consider Quantum Electrodynamics. This is only relevant to very high energies on very short scale. Like collisions between nuclei. There, you need QED to describe electromagnetic interactions properly.

When you dig deep enough in both, QED and General Relativity are built on the same principles. GR isn't quantized, but if you just look at the underlying field theory, they are the same. The difference is that GR deals with symmetries due to translation and rotation in space and time. As consequence, it can be interpreted as curvature of space-time. Electrodynamics has to do with different symmetries. So while they can be described as curving of some sort of space, it's not a very useful description.

[cantab]

However, what if I was to accelerate away to 60% of the speed of light, and the original ship was to do the same thing in the opposite direction?

Each ship would measure the other as receding at less than the speed of light. Time and space appear to distort to ensure this is the case.

-Gravity can be described as a warping of spacetime by massive objects, resulting in a force. Do other forces work this way? Can the electromagnetic force be described as a warping of some all-pervasive field, or is it sufficient to describe it more "conventionally"?

The Kaluza-Klein theory sought to describe electromagnetism in this way, employing an additional spatial dimension to do so, but it was never fully successful.

In the classical view, gravity is unique in that the force acts on the same thing that "resists" its action - mass. Therefore all objects are affected equally by gravity, and so the relativistic description of it as a distortion of spacetime works.

[Heimdall5008]

A simple way to think of it is that if you have a very, very fast baseball fly past you, from perspective of the baseball it's you who is moving near speed of light. Would your trajectory be affected by a baseball if you're flying at near speed of light by it? No. It's just a baseball. So neither would a very fast baseball have any effect on you.

I know it's a bit off topic, but as you already mentioned the relativistic baseball...

Relativistic baseball on XKCD

[Z-Man]

Would your trajectory be affected by a baseball if you're flying at near speed of light by it? No. It's just a baseball. So neither would a very fast baseball have any effect on you.

To clarify a potential misunderstanding (I read that wrong myself first): Of course a baseball passing by at .9999c will have a much bigger gravitational effect on you than one passing by at .99c. However, the effect is still tiny, and no matter how much energy you pump into the ball, the effect will be weaker than what you'd get if you convert the energy into mass and just place it next to you. The speeding ball has the severe disadvantage of only being close to you for a tiny fraction of a millisecond.

[K^2]

To clarify a potential misunderstanding (I read that wrong myself first): Of course a baseball passing by at .9999c will have a much bigger gravitational effect on you than one passing by at .99c. However, the effect is still tiny, and no matter how much energy you pump into the ball, the effect will be weaker than what you'd get if you convert the energy into mass and just place it next to you. The speeding ball has the severe disadvantage of only being close to you for a tiny fraction of a millisecond.

Actually, the net gravitational effect will be weaker at .9999c than at .99c. Has to do with factor of 2 for gravitational lensing vs classical photon trajectory.

The Kaluza-Klein theory sought to describe electromagnetism in this way, employing an additional spatial dimension to do so, but it was never fully successful.

When they started it, they were flying blind. The critical element is Yang-Mills Theory, which wasn't developed until 1950s. If you start with U(1) symmetry and apply Yang-Mills, you can build a pretty good topological model of electromagnetism, which you can expand into a differential geometric interpretation. And KK model was eventually reconsiled with that. But yeah, it ends up not very practical anyways.

[Z-Man]

Actually, the net gravitational effect will be weaker at .9999c than at .99c. Has to do with factor of 2 for gravitational lensing vs classical photon trajectory.

I think you're not doing your frame transformations properly. Let's look at it, replacing the baseball with a star. All quantities are to be interpreted as measured far away from the star where you can assume space is sufficiently flat and vector transport around the star is possible. Three objects pass the star with identical initial travel direction and projected distance to the star:

A) a spaceship at the speed of .99c

a spaceship at the speed of .9999c

C) a photon

Now, of course, the following is true: A will receive a bigger direction change than B and B will receive a bigger direction change than C. However, both spaceships are sufficiently fast that the difference is negligible. In regular situations (weak gravity fields guarantee that), trajectories depend smoothly on the initial 3-velocity, after all. If the photon is going to be deflected by 1 degree, A is not going to be deflected by 2 or 3 degrees, rather maybe 1.02 a most. In the reference frame of the star, the 3-velocity change is going to be similar in all cases, and certainly, the deflection the photon receives is a lower bound for the spaceships.

Math time. Notation: lower case letters are scalar values, upper case letters are vectors, 3 or 4 components by context. c = 1. In case there is a stray u, I mean v. Let's say before the encounter, the 3-velocity of a spaceship in the star's rest frame is is V = (v, 0, 0) and after the encounter, it's V' = (sqrt(v² - ε²), ε, 0). They have the same length, so the same É£-value of É£ = 1/sqrt(1-v²). ε is small compared to v, but has a lower bound.

Those 3-velocities correspond to the 4-velocities U = (1, v, 0, 0)É£ and U' = (1, sqrt(v² - ε²), ε, 0)É£, still in the star's reference frame.

The Lorentz transformation matrix from the star's rest frame to the rest frame of the spaceship before the encounter is (Sorry, can't write this better on the forum)

 ɣ , -ɣv, 0, 0
-ɣv,  ɣ , 0, 0
 0 ,  0 , 1, 0
 0 ,  0 , 0, 1

Easy to check by pushing U through it and getting out (1,0,0,0). What does it do to U'? It gets transformed to

( É£²(1-v sqrt(v² - ε²) ),

É£²(-v + v sqrt(v² - ε²) ),

ɣε,

0 )

and if you apply the small epsilon approximation to turn sqrt(v² - ε²) to v - ε²/2v, this simplifies to

U'' = ( 1 + É£²ÃŽÂµ²/2, -É£² ε²/2v, ɣε, 0)

What we have calculated here is an approximation of the 4-velocity of a spaceship that was initially at rest, but then is passed by a star moving by at relativistic speed -v at some predefined distance. And clearly, the change from (1,0,0,0) grows with ɣ, which grows with the energy you put into the moving object. The term dominating under normal conditions is the ɣε jerk in y-direction. If you check where the values come from, ε is proportional to the star's rest mass, so ɣε is proportional to the star's total energy measured from the initial rest frame of the spaceship.

Heck, you're not even just receiving a jerk towards the passing object of magnitude ɣε. At high (very, very high) enough ɣ values, the star starts to significantly pull you along with it.

...

Huh. At about É£ > 2/ε², the spaceship is even going faster than the star after the encounter. I did not expect that. Yeah, it's essentially just a gravity slingshot, but I would not have thought it was possible to transform an infinitesimally small deflection into one. The OP is of course right, this IS weird And entirely possible I screwed up at some point.