Suppose you have a massive black hole, with an orbiting planet. At the right distance, a planet could avoid falling into the black hole by orbiting at a sufficient high speed. As the orbit distance decreases, the orbital velocity increases-could it eventually exceed c?
Normal rules do not apply around singularities, that is why they are singularities.
Having said that I don’t see how it could, since that would indicate one of the parameters had been set to infinity.
Better suited for Great Questions perhaps? More of a physics question than a debate.
The planet can’t attain c without having an infinite amount of energy applied to it, but if you’re saying “is there such a distance at which there would be no stable orbit, since the orbital velocity would need to exceed c?” then I don’t see why not.
According to my back of the envelope calculations*, the required orbital velocity y at the event horizon of the black hole is equal to c. So any orbit that would require a velocity greater than c, would be inside the event horizon and so would effectively be part of the black hole.
- This was based on a classical mechanics modeling of the orbit dynamics, I’m not sure whether this changes when Special relativity comes in.
By the time it reached c, it would have been sucked into the black hole. Which would take infinite time (from an outsider’s POV).
Regards,
Shodan
There is a distance at which photons can orbit in a circle around a black hole, but you wouldn’t be able to get anything with nonzero mass into that orbit. Even in the vicinity of a black hole, matter still obeys the speed limit.
More GQ, really.
There is the photon sphere, which is the radius of the orbit at which a photon would make a circular orbit. Inside the photon sphere, centrifugal force actually points inwards, so it is impossible to have a stable orbit there.
Oh, and
This is correct. That distance at which photons orbit in a circle is well outside the event horizon, and there is no stable orbit contained within that distance. It’s still possible, however, to dip briefly within that region, and then zoom back out.
By ‘c’ I am assuming you are referring to the speed of light. You state that the planet would be in a ‘stable’ orbit, which means that the orbit is not decaying. Therefore, the planet would not accelerate in its orbit. But such a situation is very unlikely, I would think, so the more likely scenario is a planet falling into the black hole, and I have always wondered what would happen to something being pulled into a black hole. Especially something which was traveling at a good portion of the speed of light to begin with.
But I have been told that the energy required to accelerate mass to light speeds is infinite, because the closer a mass gets to the velocity of light, the more massive it becomes.(?) We can accelerate particles to near light speeds, but not all the way, from what I understand. Imagining a planet traveling at 90 % of the speed of light is pretty difficult for me, as the time dilation would have tremendous effects.
That’s an outdated and unhelpful description, which unfortunately has become enshrined in most books for nonspecialists. To explain: For objects at ordinary speeds, the (approximate) formula for momentum is p = mv (this is only an approximation because you can never be completely rid of the relativistic effects). This is what Newton found, and what people had gotten used to in the centuries since. Well, in relativity, it turns out that the actual formula is p = mgamma*v, where gamma is the relativistic dilation factor gamma = 1/sqrt(1-v^2/c^2). When v is much less than c, gamma is very close to 1, and so we get the approximate form we’re familiar with, but as v approaches c, gamma approaches infinity.
But people wanted to keep using a formula that looked like p = m*v, so they redefined the variables to make that happen. If you re-define mass to mean something called “relativistic mass”, which is equal to the rest mass times gamma, then you recover the old formula. But there’s no particularly good reason to do this: For purposes of most other things that depend on mass, the relevant quantity is still the rest mass. Plus, we already have a perfectly good term for “relativistic mass”, in those situations where it is useful (to within factors of c, which people don’t usually care about, since they usually use units where c = 1): That’s just the energy.
If you really insist on changing definitions, it makes much more sense to attach the gamma to the velocity instead: You define something called the proper velocity, u = gammav, and then you have that p = mu. Not only does this make more sense from first principles (gamma relates to space and time, and you’re attaching it to the part of the equation that deals with space and time), but it turns out that proper velocity is also a quite useful thing to deal with in a number of other contexts.
So, to sum up, when an actual relativist refers to “mass”, we mean the same thing that people mean when they’re not talking about relativity, the “rest mass” which doesn’t vary with speed at all. But due to unfortunate pedagogical history, when a non-relativist is talking about relativity and refers to “mass”, who knows what they mean.
I just attended a lecture by Chris Hagen, the physicist who got jobbed out of winning the Nobel prize for postulating the Higgs particle. (Which he refuses to call “Higgs”.)
His description of the way the LHC works was that magnets kept increasing the speed of the proton bundles until they couldn’t get any faster and after that all the energy went into increasing the mass. This was a popular lecture, of course.
Sigh. Yes, I’ve heard that explanation. No, I don’t know why it’s repeated by people who certainly know better. All I can guess is that they think that’s what the public wants to hear, and are sufficiently bad at communication that they can’t come up with how to describe the situation correctly.
Sadly, that absolutely describes Hagen.
Another reason why the “you can’t go faster than c because you keep getting more massive until you can’t accelerate any more” description fails is because it only makes sense if something external is giving the “push”.
If you accelerate a tennis ball by giving it a thwack with a tennis racket as it goes past, you would notice that for balls that were moving faster to begin with, the same amount of thwack is less effective at increasing the speed of the ball than it was for slower balls. Given that, it might be reasonable to imagine that the faster tennis balls are more massive.
But if you’re in a spaceship being accelerated by a rocket engine, the rocket is always at rest with respect to the spaceship, so should only see the spaceship as having its rest mass. So why should the rocket be less effective at accelerating when the spaceship is going “fast”. The “stuff gets more massive when it goes fast” model can’t give a reason.
In the Large Hadron Collider there actually is an outside source giving the protons additional energy.
Which is why for the large hadron collider, it’s more defensible to use the “can’t get faster than c because the protons keep getting more massive” explanation.
But the “can’t get faster than c” result also applies to rockets, and it is much harder to apply the “getting more massive” reason to them, because their engines are always at rest relative to the engine that is providing the motive force. Why should it matter that the rocket is getting “more massive” as seen in some distant rest frame?
And in fact, in a rocket’s own reference frame, it can indeed keep on accelerating indefinitely (well, at least until it runs out of fuel).