Accel Qs: Is Velocity Affected by Gravitational Acceleration?

[Rlam90]

Now, this doesn't exactly pertain only to the theories of relativity, but I've been recently curious how this really works.

I know that the gravitational acceleration at a point around a mass is Gm/r^2. This means that at each point in a 1-dimensional distance from the center of a mass has a different acceleration. Basic logic then tells me that as an object actually undergoes this acceleration towards the center of the mass, the rate of acceleration increases. I just want someone to confirm that the total distance traveled and velocity at any given time is, in fact, influenced by this.

This would bring up an interesting occurrence where the velocity of an object increases at a continuously faster rate until it reaches a point very close to the center of mass, where it would slow back down and even reverse. I'm assuming this is how (at least on a very minute scale) objects in a total vacuum universe would react. Any arguments?

 

[mathman]

Your question is a little confusing, especially toward the end. Is the attracting mass a point or is it extended? Is the object being pulled in penetrating the attracting mass?

 

[pervect]

Also, is this a question about Newtonian gravity or GR? Since it's in the GR forum, I'm guessing the late - but your starting point, and the entire framework of your question, is Newtonian.

 

[ghwellsjr]

The answer to your question is yes and that is why orbiting bodies follow elliptical paths where the velocity increases as the orbiting body approaches the massive body and then it decreases as it finally moves away from its closest approach to the massive body.

If you thought that up on your own, I would say that you are very brilliant.

 

[Rlam90]

Heh. I believe I'm having some difficulty in communicating in a field in which I rarely get the chance to have anything more than a monologue in. Plus I was a little tipsy that night.

I think my real point was to ask for the real solution for a hypothetical particle traveling in a 1-dimensional line toward the gravitating object. I'm sure I could work it out, but I don't have any formal calculus training nor any interaction with anyone who knows it, so progress would probably be pretty slow.

Anyone understand this request? Or do I just need to earn my boyscout merits before I start using this forum to increase my understanding of how this universe puts itself together?

Oh, and just to clear something up:

If you thought that up on your own, I would say that you are very brilliant.

Do I smell the scent of sarcasm here, or am I touched with a bit of paranoia? I know everyone tells me this in person, but in person you rarely encounter people of the same caliber of intellect as one would expect you to find on a forum dedicated to such a mentally taxing subject as physics.

I made a mistake in punctuation while originally writing this response; this leads me to believe the former.

 

[ghwellsjr]

If those are the only two choices--well, the first one isn't correct--I was being totally sincere. Maybe you need to learn how to receive a compliment, especially when you don't think it is deserved.

Let me say this about your question. In physics, one way to treat large gravitational masses is to treat them as particles, that is, bodies with no dimensions, in other words points. (See the wikipedia article on Point Particle.) The idea is that you are describing the motion of the center of mass of the bodies. Now one of the bodies can be very large which we consider to remain stationary and the other one can be very small which we consider to be free to move. If you image two such particles that are separated by some large distance but otherwise stationary with respect to each other and then you "drop" the smaller one, it will accelerate toward the larger one, ever increasing in speed until it passes right through the "larger" stationary one and then decelerates on the other side until it achieves the same distance on the other side as it had when it started.

But it sounds like you are already aware of all this. Were you looking for a formula?

 

[Passionflower]

First you should specify what you are looking for. Are you looking for an answer using Newtonian theory or general relativity?

 

[Rlam90]

If those are the only two choices--well, the first one isn't correct--I was being totally sincere. Maybe you need to learn how to receive a compliment, especially when you don't think it is deserved.

Let me say this about your question. In physics, one way to treat large gravitational masses is to treat them as particles, that is, bodies with no dimensions, in other words points. (See the wikipedia article on Point Particle.) The idea is that you are describing the motion of the center of mass of the bodies. Now one of the bodies can be very large which we consider to remain stationary and the other one can be very small which we consider to be free to move. If you image two such particles that are separated by some large distance but otherwise stationary with respect to each other and then you "drop" the smaller one, it will accelerate toward the larger one, ever increasing in speed until it passes right through the "larger" stationary one and then decelerates on the other side until it achieves the same distance on the other side as it had when it started.

But it sounds like you are already aware of all this. Were you looking for a formula?

Yeah, that's what I was looking for. And I was thinking more along the lines of the point of reference being the larger particle, where this particle remains completely stationary in a coordinate system and only the smaller object is mapped with a trajectory relative to the larger. I was hoping for a simple equation that would map the trajectory on a one-dimensional line, such as v=(Gm/r^2)t, but taking the decreasing distance into account. I don't so much care about the rigorous formulas that would be involved in 2- or 3-dimensional movement, but rather only care about the simple 1-dimensional time-to-distance function. I'm hoping this is a relatively simple request, but as I lack a complete understanding of what the solution may be comprised of, I may be wrong. I haven't even tried to figure it out with calculus. I'm sure this would be the easiest way of going about this.

It would also be useful to me if somebody could take the lorentz factors into account for such a time-to-distance function if that is possible in a simple manner. I'm interested in finding a tool to simplify mental simulations and experiments, and as I know the effects of relativity come into account in almost every situation involving gravity, I'd like to account for these. Optimally I'd like to see an equation that would account for the distance traveled taking the time slowing from the smaller object accelerating relative to the stationary larger object's frame of reference, as well as from the larger object's gravitational field.

After I practice using these ideas, I'm planning on incorporating the rotational properties of these particles into effect, as well as their electromagnetic and other properties. It seems like I appear to be thinking in Newtonian fashion, but its simply to most convenient frame of thought for me to conduct thought experiments in with my current knowledge. I have no problem incorporating further effects from relativity and even quantum mechanics into account, apart from a difficulty interpreting the language involved in communicating the ideas in these theories.

And I apologize for not knowing how to take a compliment. This is a different environment than I am used to, and I have been exposed to forums such as this that did have a very disturbing rate of elitism in them. I'm relieved to see that I am being received in a much more mature manner than I have been in the past. Also note that I am only 2 days over twenty years old and I have only taken one semester of classes since I dropped out of high school in 2008.

 

[pervect]

We've had a few threads on that. Without calculus, though, it will be hard to do anything other than give the barest results from the plethora of recent threads on the topic.

The Schwarzschild radial coordinate isn't really a distance in GR as it is in Newtonian physics, which complicates matters. It's a generalized coordinate - it describes the location of the particle, but subtracting the R values does not necessarily have any physical significance as a "distance" - though under the right circumstances, it can come surprisingly close - (close, but no cigar). (This was one of the topics of the recent debate, I really hope we're all agreed on that, though I'm probably being optimistic. If we're not agreed on it, I hope we're at least tired of debating the point).

The easiest formula for the R coordinate as a function of time is in terms of proper time. Proper time is the time that your infalling particle would measure on his or her wristwatch, if it was wearing one.

Introducing coordinate time gives very complicated formulas, which would be needed if you wanted to do more advanced things, but I'm guessing you don't. A complete specification of the path of the particle requires specifying R(tau), the schwarzschild R coordinate as a function of proper time, and T(tau), the schwarzschild T coordinate as a function of proper time, however.

A simple equation for an object dropped from infinity into a black hole with a Schwarzschild radius of one unit is

r(tau) = (-3/2 tau) ^ (2/3)

Tau is a negative number, when tau = - infinity, the object is at infinity, when tau = 0, the object is at r=0 and is destroyed.

There are a couple of ways one might measure velocity, the physically meaningful one is the velocity of the infalling observer relative to someone stationary at various R values. For the fall from infinity, this would be c/sqrt(r). So at r=infinity, the velocity is zero, as r approaches the event horizon, at r=1, the velocity approaches the speed of light.

You might expect that the velocity would be dR/dT (or maybe without calculus, you wouldn't). Because R and T are generalized coordinates, this isn't the sort of velocity anyone would actually measure, however. And because of the way the Schwarzschild T coordinate is defined, dR/dT goes to zero as the object approaches the event horizon, rather than rising.

The short explanation for the difference here is that besides from the issue of the R coordinate not being precisely a distance, the notion of time of the wristwatch time carried observer (proper time), the notion of time of the observer at infinity, and the notion of time of an observer at an arbitrary stationary R value are all different.