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Suppose you are at rest relative to a black hole (so you are maintaining some constant acceleration in order to oppose that gravitational attraction), outside the event horizon. Now a continuous stream of energy starts falling into the black hole. This will cause the event horizon to grow and the gravitational potential of the black hole to increase, meaning that you would begin to fall toward the event horizon (because your acceleration is no longer sufficient to oppose the gravitational potential).

Now suppose that rather than the black hole growing, it remains at a constant size (nothing is falling in), and the observer's acceleration (which is opposing the gravitational force) decreases (his rocket is losing power, but he is unaware and unable to measure this power - I'm thinking along the lines of Einstein's equivalence principle). Again he begins to fall toward the black hole which will seem to him to be increasing in size.

Is there any way for the observer to tell the difference between these two cases (i.e. is there a way for the observer to know that the decrease in distance between him and the black hole is due to an increase of mass in the black hole or a decrease in his own acceleration)?

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Assume the rocket is hovering above the hole.

If the rocket loses thrust, an accelerometer on the rocket will measure less acceleration. Thus, an astronaut will feel less heavy.

However, if the gravity of the hole increases, the accelerometer reading will not change; the astronaut will not feel any change in weight.

But, in both cases, the rocket will begin falling toward the hole.

So, yes, the observer can tell the difference between the two cases.

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  • $\begingroup$ But in both cases, the observer would be in the same type of freefall, why would an accelerometer measure a difference (how can the accelerometer tell the difference between the causes of the acceleration toward the black hole)? $\endgroup$ – Chris L. Nov 22 '13 at 0:44
  • $\begingroup$ @PetTaxi, in neither case is the observer in free fall. Only if the rocket engine is shut-off will the rocket be in free-fall. Free-fall means free from all force (recall that, in GR, gravity is not a force). In free-fall, the accelerometer reads zero; the astronaut feels weightless. I think you've got some homework to do. $\endgroup$ – Alfred Centauri Nov 22 '13 at 0:55

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