Traveling at the speed of light in orbit around a black hole

[Somedude99]

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[bcrowell]

Stephen Hawking says that orbiting around a black hole would slow down time for those in orbit because of the incredibly large mass.

A black hole does not have to have an incredibly large mass. It's not a large mass that makes a black hole, it's a large density.

According to Hawking time would be cut down by half when in orbit.

Why half? Are you just using that as an example?

What happens? If light is constant, from Earth it would look like it's only orbiting at the speed of light, but in reality it should be going faster.

Here I assume you mean "near the speed of light," rather than "at the speed of light."

I don't think anything in your example really requires a circular orbit. All the same ideas apply if you just let the space traveler A fall straight towards the black hole.

To an observer B hovering a millimeter outside the event horizon, A passes by at very nearly the speed of light. But to a distant observer C on Earth, A appears to be moving more and more slowly as he approaches the hotizon, due to gravitational time dilation: clocks deep in a gravity well run more slowly. So the effect as seen by C is in the opposite direction compared to what you'd been thinking: A's velocity seems reduced, not increased.

 

[Somedude99]

I deleted the message because I thought I would find out myself. Since you replied already I'll try to explain further.

If the ship at the black hole appears from Earth to be moving slower in time... then if they were traveling at the speed of light how would they look from earth? The black hole is slowing down time but they are still traveling at the speed of light, so how do they end up?

 

[Somedude99]

In other words, the ship in orbit should be traveling at near the speed of light, viewable from earth. However, the black hole slows down their time so they should be going slower. But how can light be going slower? How can time distort if two different perspectives view time differently?

 

[JesseM]

I deleted the message because I thought I would find out myself. Since you replied already I'll try to explain further.

If the ship at the black hole appears from Earth to be moving slower in time... then if they were traveling at the speed of light how would they look from earth? The black hole is slowing down time but they are still traveling at the speed of light, so how do they end up?

It's impossible for an object with mass to travel at the speed of light, but you could imagine the ship traveling at some large fraction of the speed of light, say 0.99999c. But note that even in the absence of gravity, if something is traveling at a large fraction of light speed their clocks will be slowed down in your rest frame, not sped up...so in the case of a black hole, the gravitational time dilation would just slow down their clock even further as measured by distant observers.

edit: Maybe you're not talking about clock rates at all, but just the speed an object appears to be moving? Note that in relativity the "speed" of anything depends on your choice of coordinate system for defining the position and time coordinates of different events (with "speed" as change in position coordinate/change in time coordinate)--light is only guaranteed to have a speed of c in a special class of coordinate systems called inertial coordinate systems, the type of coordinate system used in a large region of curved spacetime like around a black hole will be a non-inertial one, since inertial systems can only be defined in large regions in non-curved spacetimes (no gravity). However, by the equivalence principle, if you pick a small region of a larger spacetime, a coordinate system constructed out of free-falling rulers and clocks will measure something close to what would be measured in an inertial coordinate system, including the fact that light beams in this region would move at c; so in this sense, it is still true that light moves at c in any "local region" of a curved spacetime like the spacetime around a black hole.

 

[bcrowell]

In other words, the ship in orbit should be traveling at near the speed of light, viewable from earth.

Nope. You're assuming this, but it's not true. See #2.

 

[Passionflower]

The speed of light measured at a point is always c, in all frames and all spacetimes.

But if we go beyond that and measure the average speed between two points there are three situations to consider:

An inertial frame in flat spacetime (curved spacetimes have no inertial frames)
In this frame the speed of light is c in all situations.

An accelerating frame in flat spacetime
In this frame the speed of light is not c.

In curved spacetime
The speed of light is not c.