Thread: Thought experiment with two contradictory consequences

Imagine a particle (a photon, say) travelling towards an observer, standing on a planet (or a neutron star or anything heavy). The photon has an extremely high frequency, as seen in the frame of the observer. Actually, it has so much energy, that it’s just about collapsing into a black hole. As the photon gets deeper down the gravity well of the planet/neutron star, it gets blueshifted, which increases the energy, and the photon turns into a black hole.

Another observer lives on another planet, which is travelling towards the first observer at a significant speed. In the frame of this observer, the photon isn’t quite as close to collapse into a black hole. In fact, the blue shift of the photon caused by the gravity well of the first observers planet isn't enough to transform the photon into a black hole.

Now, both observers have to be right. But how can it be, that the photon is and isn’t collapsing into a black hole? It seems that the photon has two histories, which sometimes can be acceptable. But in this case, imagine that the first observer gets swallowed by the newly formed black hole. Then the second observer can travel to the first observer’s planet and find out, which make a serious contradiction.


What is the solution of this thought experiment?

The photon does not collapse into a black hole for any frame. The assumption that it would according to the first observer is incorrect.

Indeed. Photons can't collapse into black holes because they don't have mass. For that matter, photons don't really have a size either, so its not like it can collapse into anything.

Originally Posted by jmd_dk

What is the solution of this thought experiment?

It's simpler than you think: as the photon falls into the black hole, it doesn't gain any energy. Conservation of energy applies.

As pointed out by Janus and Xellos, there are some issues with using a photon for this thought experiment, so let's consider "a mass" in a less extreme situation. Consider a 1kg cannonball in free space. It falls to Earth with considerable kinetic energy and a final velocity of circa 11km/s. Where did this kinetic energy come from? People tend to say "from the gravitational field", but that's wrong. To understand why, reverse this scenario. When you lift the cannonball you do work on it, giving it gravitational potential energy. When give so much lift that you've given it gravitational potential energy equivalent to the 11km/s worth of kinetic energy, it ends up in free space where the Earth's gravity is negligible. Conservation of energy tells you that the energy you expended is now in the cannonball. It hasn't gone into the Earth's gravitational field, instead that's reduced a little because the departing cannonball has removed energy from the system. Check out binding energy and in particular the mass deficit section. You have "unbound the system", and as a result the mass of the cannonball has now increased. That's where the gravitational potential energy has gone, and that's where the kinetic energy of a falling cannonball comes from. From the cannonball.

Now go back to the photon. Where does the apparent blue-shift energy-gain come from? From the photon. Which means it hasn't gained any energy at all. It only appears to have done so because down near the event horizon, an observer is subject to gravitational time-dilation. So he measures the photon frequency as being higher than the observer out in free space.

This is the important point about gravity when it comes to general relativity. It's not a force in the mechanical sense. When you push a cannonball you do work on it, you exert force over a distance, you accelerate it, you give it kinetic energy. In general relativity, the principle of equivalence relates this acceleration to a body resting on the ground, not to a body falling to earth. The falling body isn't being "accelerated", there is no mechanical force acting upon it, and it isn't gaining any energy from any outside source. Conservation of energy applies.

Originally Posted by Xelloss

Indeed. Photons can't collapse into black holes because they don't have mass. For that matter, photons don't really have a size either, so its not like it can collapse into anything.

I see that both you and Janus says that photons can't collapse into black holes. You'r explanation as to why though, is incorrect. Photons have energy, which is the crucial point (energy curves spacetime in exactly the same way as mass). Also, photons have a size just as well defined as any other particle. Also, no photon with a wavelength smaller than the Planck length can exist, as this would create a black hole, because the photon would have a very large amount of energy in a sufficiently small space. That is facts, which I don't want to argue about.
You/Janus could be right that no photon can ever collapse into a black hole, because it can never reach the required smallness of the wavelength of just 1 Planck length. If that's so, please tell me why this cannot happen?

Originally Posted by Farsight

Now go back to the photon. Where does the apparent blue-shift energy-gain come from? From the photon. Which means it hasn't gained any energy at all. It only appears to have done so because down near the event horizon, an observer is subject to gravitational time-dilation. So he measures the photon frequency as being higher than the observer out in free space.

I need you to clear a few things out. What do you mean, apparent blue-shift? Also, you agree that although the total energy of the photon hasn't changed, the observer near the event horizon will see it with a smaller wavelength than the other observer. Can this wavelength be smaller than the Planck length? If so, why doesn't it collapse into a black hole? And if not, what prevents the photon from getting"too" blue-shifted?

Originally Posted by jmd_dk

Originally Posted by Xelloss

Indeed. Photons can't collapse into black holes because they don't have mass. For that matter, photons don't really have a size either, so its not like it can collapse into anything.

I see that both you and Janus says that photons can't collapse into black holes. You'r explanation as to why though, is incorrect. Photons have energy, which is the crucial point (energy curves spacetime in exactly the same way as mass). Also, photons have a size just as well defined as any other particle. Also, no photon with a wavelength smaller than the Planck length can exist, as this would create a black hole, because the photon would have a very large amount of energy in a sufficiently small space. That is facts, which I don't want to argue about.
You/Janus could be right that no photon can ever collapse into a black hole, because it can never reach the required smallness of the wavelength of just 1 Planck length. If that's so, please tell me why this cannot happen?

That actually has nothing to do with it. No particle, even one with a non-zero rest mass (such as an electron) would collapse into a black hole due to an increase in its relativistic mass.

The reason is the very basis of your supposed "contradiction". Since the particle is at rest in its own frame, It's only measured "mass" is it's rest mass, since this is not enough to collapse it into a black hole, it does not do so in its own rest frame. And if it does not collapse into a black hole in its own frame, it cannot collapse into one according to any other frame either.

Put another way, If you properly apply GR to the situation an observer watching said particle traveling by at some high fraction of c, you will find that it does not not predict that the particle will form a black hole due to the increase of its relativistic mass as seen by that observer. This is due to the way that gravity fields transform between frames with relative motion with respect to each other.

Originally Posted by jmd_dk

I need you to clear a few things out. What do you mean, apparent blue-shift? Also, you agree that although the total energy of the photon hasn't changed, the observer near the event horizon will see it with a smaller wavelength than the other observer. Can this wavelength be smaller than the Planck length? If so, why doesn't it collapse into a black hole? And if not, what prevents the photon from getting"too" blue-shifted?

I'd have to explain photons in some detail to explain this, and plumb the depths of electromagnetism and talk about displacement current. It would take me a long time, too long.

Instead think about a really massive star. You accelerate towards it, and it appears to have gained an enormous amount of kinetic energy. But that's just how it appears to you. It hasn't actually gained any energy at all, not really. It isn't going to start collapsing into a black hole just because you moved towards it. It's the same if you move towards a photon. You see it as blue-shifted, it appears to have gained energy. But the photon hasn't changed, you have. The same kind of thing happens when you go to a place where gravitational potential is lower. You change, you see the photon as blue shifted, but it hasn't really gained any energy.

Originally Posted by Farsight

Instead think about a really massive star. You accelerate towards it, and it appears to have gained an enormous amount of kinetic energy. But that's just how it appears to you. It hasn't actually gained any energy at all, not really. It isn't going to start collapsing into a black hole just because you moved towards it. It's the same if you move towards a photon. You see it as blue-shifted, it appears to have gained energy. But the photon hasn't changed, you have. The same kind of thing happens when you go to a place where gravitational potential is lower. You change, you see the photon as blue shifted, but it hasn't really gained any energy.

Except in the latter scenario, the photon has changed: it is in a location with a higher gravity.

PhysBang: regardless of any subtle changes to the falling photon, it really doesn't gain any energy. See my earlier post. Conservation of energy applies to the cannonball, and to the photon.