Acceleration

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Acceleration

#1  Postby twistor59 » Nov 22, 2011 8:32 am

Re. this question from another thread:

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.
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Re: Acceleration

#2  Postby starkiller » Dec 08, 2011 12:44 am

Yep, I got confused and I think i got it now.

The next question I have is.
Does this apply to a photon?

Depending on if the answer is yes, then the next question would be(hypothetical one of course)
What would happen to a particle that is traveling at c which passes by a blackhole within it's gravitational field right at the outer edge of photon sphere. Would it travel faster than C since it is traveling at C speed before it's affected by the gravitation field?

same question for a photon.
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Re: Acceleration

#3  Postby twistor59 » Dec 08, 2011 9:14 am

starkiller wrote:Yep, I got confused and I think i got it now.

The next question I have is.
Does this apply to a photon?

Depending on if the answer is yes, then the next question would be(hypothetical one of course)
What would happen to a particle that is traveling at c which passes by a blackhole within it's gravitational field right at the outer edge of photon sphere. Would it travel faster than C since it is traveling at C speed before it's affected by the gravitation field?

same question for a photon.


You have to be very careful about what is meant with the term "speed of the photon". In flat space, you can choose a time coordinate and a distance coordinate and calculate the speed as (x2-x1)/(t2-t1). You get the famous answer "c". But of course this calculation might depend on the distance and time coordinates you chose. The key thing about relativity is that it doesn't depend on the coordinates you choose, provided you restrict yourself to a bunch of coordinates corresponding to inertial frames. The coordinates of the various possible inertial frames are related to each other by Lorentz transformations.

Now if I choose some other coordinates, say u and v which aren't related to t and x by a Lorentz transformation (maybe they're wiggly functions of t and x with powers, sines and cosines etc), then if I compute the speed of a photon as (v2-v1)/(u2-u1), I can get different answers, not just c any more !

Now in curved space near your black hole, it's impossible to find inertial coordinates, except, if I take a point, then in an infinitesimally tiny region around that point, I could define them. So I'm stuck with arbitrary coordinates in general. Hence dividing distance differences by time differences in these coordinates gives me arbitrary values for the speed of the photon,
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Re: Acceleration

#4  Postby starkiller » Dec 08, 2011 9:52 pm

I think from that paragraph the simple answer might be that it might be possible, but we don't have the means to know for sure.
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Re: Acceleration

#5  Postby starkiller » Dec 10, 2011 4:14 am

For some reason I can't edit that, but it should have a ? at the end.
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Re: Acceleration

#6  Postby twistor59 » Dec 10, 2011 9:59 am

starkiller wrote: Would it travel faster than C since it is traveling at C speed before it's affected by the gravitation field?

same question for a photon.


The key point is the bit I highlighted. In order to answer the question, you have to say exactly how you're going to define and measure the speed. The short answer is that if you measure it in a local inertial frame, then no, the speed must be less than c, and for a photon, the speed will be c.

Even in flat space - no gravity, no black holes, I can choose an accelerated frame (non inertial) in which the speed is > c.
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Re: Acceleration

#7  Postby starkiller » Dec 15, 2011 1:19 am

twistor59 wrote:
starkiller wrote: Would it travel faster than C since it is traveling at C speed before it's affected by the gravitation field?

same question for a photon.


The key point is the bit I highlighted. In order to answer the question, you have to say exactly how you're going to define and measure the speed. The short answer is that if you measure it in a local inertial frame, then no, the speed must be less than c, and for a photon, the speed will be c.


Ok. Thats where the question and the answer do not make sense to me. If the Speed of the photon is C and it's accelerated by gravity what is the speed? my answer would be >C or else how could a photon traveling C be accelerated.
Last edited by starkiller on Dec 15, 2011 1:45 am, edited 1 time in total.
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Re: Acceleration

#8  Postby starkiller » Dec 15, 2011 1:44 am

twistor59 wrote:
Even in flat space - no gravity, no black holes, I can choose an accelerated frame (non inertial) in which the speed is > c.

So then the photons in that frame when they measure themselves do not see themselves traveling faster than C, but they are actually traveling faster than C if you measure from outside that frame?

If that is the case then if the photons traveling c in-frame are accelerated via gravity to >c and the frame is accelerated >c then the photons are traveling (i+o)>c (i=inframe,o=outframe)?

Or maybe i missed the boat completely.
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Re: Acceleration

#9  Postby twistor59 » Dec 15, 2011 8:58 am

The question can't be answered using the terminology you're employing. In setting up these sorts of questions you have to be ultra-precise. Even a term like "speed" is way too ambiguous. For an example of what I mean, and how even experienced people can misunderstand each other, see the discussion here.
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Re: Acceleration

#10  Postby starkiller » Dec 15, 2011 6:07 pm

twistor59 wrote:The question can't be answered using the terminology you're employing. In setting up these sorts of questions you have to be ultra-precise. Even a term like "speed" is way too ambiguous. For an example of what I mean, and how even experienced people can misunderstand each other, see the discussion here.


i read it and that made me have more questions, esp about the emitter and tower part. OMG.
And the special formula we are "supposed to use". My first question to that is Why?

What term should be used instead of speed?
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Re: Acceleration

#11  Postby twistor59 » Dec 16, 2011 8:54 am

starkiller wrote:
twistor59 wrote:The question can't be answered using the terminology you're employing. In setting up these sorts of questions you have to be ultra-precise. Even a term like "speed" is way too ambiguous. For an example of what I mean, and how even experienced people can misunderstand each other, see the discussion here.


i read it and that made me have more questions, esp about the emitter and tower part. OMG.
And the special formula we are "supposed to use". My first question to that is Why?

What term should be used instead of speed?


There's nothing wrong with "speed", but it needs to be supplemented to say how you're going to measure it:

Do you just take (x2-x1)/(t2-t1) ? - average coordinate speed
Do you take a limit i.e dx/dt ? - instantaneous coordinate speed
What are the restrictions on x and t - are they arbitrary coordinates or locally inertial ?
How do you define the t coordinate ? - with a bunch of clocks all over the place - how do you synchronize the clocks ?..

Here's a reference
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Re: Acceleration

#12  Postby hackenslash » Dec 16, 2011 11:13 pm

starkiller wrote:Ok. Thats where the question and the answer do not make sense to me. If the Speed of the photon is C and it's accelerated by gravity what is the speed? my answer would be >C or else how could a photon traveling C be accelerated.


You also need to be clear on precisely what is meant by 'accelerated'. It doesn't only mean 'go faster', it describes any change in velocity, including changing direction, speeding up and slowing down.
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