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Lets' say that something is in a universe with a perfect vacuum, where nothing exists but that object. This universe has an infinite amount of space and time. Now that object emits a single photon. Will this photon eventually return to the sender due to the senders own gravity, or will the photon continue to travel straight for the rest of eternity?

Photons are affected by gravity, thats why they bend around black holes. But I've read or heard somewhere (I have no sources for this, so this could be wrong) that the speed of a photon is constant. So if the photon wants to return to the sender it would have to slow down, and eventually start going the oppesit direction, which would defy the rule that the speed of light is constant.

I can't come with a logical conclusion to this problem, so some help would be appreciated.

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  • $\begingroup$ In your infinite open universe once a photon is emitted by an object it is not coming back. The speed of light is indeed constant to all observers regardless of their speed. Photons never slow down or speed up. $\endgroup$ – StephenG Apr 12 '18 at 10:14
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Assuming your object is spherically symmetric then the geometry you are describing is the Schwarzschild metric i.e. the geometry around any spherical massive object. So we just need to calculate the trajectory of light in this geometry. This is discussed in many existing questions e.g. Speed of light in a gravitational field?

  1. If you are outside the event horizon then the ray of light travels outwards and escapes to infinity.

  2. If you are inside the black hole then ever a light ray emitted in the outward direction travels inwards and hits the singularity at the centre of the black hole.

  3. If the light ray is emitted smack bang on the event horizon then it remains stationary neither moving or out of the black hole.

For more on this see Why is a black hole black? though that might be a bit too detailed.

So the answer to your question is that the light ray never stops and doubles back on itself. If emitted radially it only ever travels in a straight line in the same direction. Since you don't mention a black hole I assume you are thinking of an object that isn't dense enough to be a black hole. In that case the option applies (1) and the light ray speeds away in a straight line never to return.

If you are the observer sitting on the surface of the object then formally the light ray leaves you at the speed of light and then increases in speed as it moves away. However in general relativity you need to be careful about what is meant by the speed of light as it's a more subtle issue than you might think. I go into this in detail in my answer to GR. Einstein's 1911 Paper: On the Influence of Gravitation on the Propagation of Light.

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Let's say that something is in a universe with a perfect vacuum, where nothing exists but that object. This universe has an infinite amount of space and time. Now that object emits a single photon. Will this photon eventually return to the sender due to the senders own gravity, or will the photon continue to travel straight for the rest of eternity?

I have to say I'm not fond of infinity and eternity here, but the answer is the latter. That photon will not "return to sender".

Photons are affected by gravity, that's why they bend around black holes.

True enough. They bend round stars too. See Professor Ned Wright's deflection and delay article: "In a very real sense, the delay experienced by light passing a massive object is responsible for the deflection of the light".

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But I've read or heard somewhere (I have no sources for this, so this could be wrong) that the speed of a photon is constant.

That's wrong. See what Einstein said: "A curvature of rays of light can only occur where the speed of light is spatially variable".

So if the photon wants to return to the sender it would have to slow down, and eventually start going the opposite direction, which would defy the rule that the speed of light is constant.

It isn't going to do that. Have a look at the physicsFAQ article Is The Speed of Light Everywhere the Same? The answer is no. Note the General Relativity section, which refers to Einstein, and says this: "light speeds up as it ascends from floor to ceiling". Your emitted photon isn't like some stone thrown upwards. It doesn't slow down. It speeds up.

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