starkiller wrote:Hello,

New to this forum and I have a question. I figured this was the post to post it in.

as a particle is approaching a black hole does it accelerate?

as a particle is approaching a massive sun does it accelerate?

I'm guessing that you mean a particle drifting in space, under the gravitational influence of the sun/black hole ?

I don't know how much experience of GR you have, so I apologize if I'm repeating stuff you already know:

In GR particles in free fall follow a "shortest" path through spacetime called a geodesic. Given a starting point (and starting velocity) for the particle, the geodesic it will follow is determined by the gravitational field existing in the region. In your case it is the gravitational field set up by the sun/BH. Now in GR, when you're moving along a geodesic, from your point of view, you're not accelerating - you don't feel any force.*

Strictly speaking the term you're looking for is proper acceleration. Essentially, the proper acceleration at a point on your trajectory expresses the difference between your motion through spacetime and the motion of the (unique) geodesic through that point with the same velocity that you have there. So if you're on a geodesic yourself, your proper acceleration will be zero.

Rather confusing, but if you're stood on a surface of a planet you have non zero proper acceleration, but if you're falling you have zero proper acceleration !

So getting back to your question, the particle falling towards the sun/BH is precisely one that is NOT accelerating (from its own point of view). Of course an extenal observer could compute the components of some derivative of the four-velocity of the particle with respect to the geodesic's parameter and call that acceleration if he wishes, but that's entirely coordinate dependent.

*Small caveat: if the gravitational field is not uniform, you will feel a tidal force due to the fact that geodesics are converging or diverging in the region you're in. So, for example, in free fall in strong gravity you may find your left testicle moving to the left and your right moving to the right. Or vice versa.