Do "light" and "heavy" objects still accelerate at the same rate at near light speeds?

Question

My (very poor) layman's understanding of physics is that as something approaches c, it has a relativistic mass that is a combination of its actual mass and velocity.

Do this affect its acceleration due to gravity?

In other words, does a given "light" object and a given "heavy" object hit a relativistic speed of e.g. (.9)c at the same time if their acceleration is entirely due to gravity?

My gut says yes, but I couldn't say why. I'm only skeptical because of the other layman's physics expression along the lines of "It takes more and more energy to accelerate something as you approach the speed of light", but I suppose that energy would come from gravitation potential energy (which is already a product of mass) in the above thought experiment.

Answer 1

Suppose you have a 1kg mass falling into a black hole. You can divide this into ten 100g masses, a hundred 10g masses and so on. If smaller masses fell at a different rate from larger masses then you'd find that just dividing an object changes its rate of fall, which would be strange to say the least.

For an object falling into a static black hole it turns out to be quite easy to calculate the fall velocity, and I discuss this in my answer to Will an object always fall at an infinite speed in a black hole? The velocity does not depend on the mass of the object.