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Acceleration from curved spacetimeby Jorjy
Tags: curved spacetime 
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#1
Feb1211, 04:05 PM

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This might be a stupid question..
Why does curved spacetime cause objects with mass to accelerate towards each other? If I placed a massive particle next to a larger massive object, at rest with respect to the large object, shouldn't the particle stay at rest? 


#2
Feb1211, 08:15 PM

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Er...
GRAVITY causes masses to accelerate towards each other. In a Newtonian framework, gravity is a FORCE, which causes objects to accelerate towards each other according to F=ma. In a General Relativistic framework, gravity is a curvature in spacetime. Particles follow trajectories in spacetime (also called worldcurves), e.g. (ct, x(t), y(t), z(t)) and these world curves are determined by the geodesics of the spacetime. You will find that the geodesics or world curves are such that massive objects are accelerated towards each other (almost the same way they are in Newtonian gravity, with larger and larger discrepancies occurring for larger and larger curvatures). 


#3
Feb1311, 08:46 AM

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Without gravity, we draw the graph on flat paper. Objects moving freely (with no force on them) have graphs that are straight lines. With gravity, we draw the graph on a curved surface. Two objects that begin at rest relative to each other have worldlines (graph lines) that are initially parallel, but the curved surface causes the lines to converge or diverge further along. See also www.relativitet.se/spacetime1.html 


#4
Feb1311, 01:18 PM

P: 6

Acceleration from curved spacetime
Thanks Matterwave and DrGreg.
DrGreg, I was thinking about curved spacetime as in the picture on the link you gave when I asked my question. I understand how a moving object follows a straight line along the curved spacetime. What I am wondering about is why an object that is placed at rest (say, 1 meter above the earth) would accelerate towards the earth (I guess both would be accelerating towards each other). For example, in the curved spacetime cone in the picture, the placed object is at rest 1 meter away from the earth, and since it is not moving, it isn't following the curvature toward earth. So why does it accelerate toward earth? Does curved spacetime cause the object to sort of slide toward earth? 


#5
Feb1311, 01:24 PM

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You're thinking about curved space, not curved spacetime. Curved space only causes moving things to accelerate. Curved spacetime is also curved (typically by approximately the same amount) with respect to time, so when you plot position in space against time in equivalent units you get the same curvature, corresponding to an acceleration.



#6
Feb1311, 01:28 PM

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In relativistic physics the word 'motion' does not mean the same thing as in classical physics. 


#7
Feb1311, 02:02 PM

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The background. When you throw an object upwards then at the point in time where it reaches its highest point it has momentarily zero spatial velocity (relative to you, the thrower). In a sense you are asking why objects, when they have reached their highest point, fall back again. www.relativitet.se/spacetime1.html What you seem to think of is the cone as an inclined surface, with a force causing objects placed on it to slide in some direction. The image of the cone does not have that purpose at all. Here the entire base of the cone represents a single point. You throw an object straight upwards, and it fall right back in your hand. The line along the surface of the cone has the following property. At every point along the way the line does not curve right or left relative to the local surface. The line curves back down because of the curvature of the surface as a whole. 


#8
Feb1311, 04:12 PM

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Your question is a good one, one I still have trouble "visualizing".
I sure don't REALLY understand that volcano shaped image above...try as I might. Isn't it easier to visualize from this diagram: http://en.wikipedia.org/wiki/Spacetime A potentially confusing aspect of spacetime is discussed here: Acceleration is perpendicular to velocity http://www.physicsforums.com/showthread.php?t=470056 


#9
Feb1311, 04:17 PM

P: 6

Yeah, I think my problem is that I'm not considering the time component of spacetime..
It's difficult for me to conceptualize curved spacetime. Even the cone is confusing to me because the time coord wraps around to meet back at the original point. Anyway, I'll keep thinking about it. Thanks for the help! 


#10
Feb1311, 04:23 PM

P: 5,632

It IS confusing...
from my link above: ... And you will note that in spacetime, velocity is constant (c)....and acceleration doesn't change that speed...it changes the shape of the worldline thru spacetime.... 


#11
Feb1311, 04:23 PM

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#12
Feb1311, 04:29 PM

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#13
Feb1311, 04:44 PM

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Dr Greg..I was afraid somebody would say that....
why is it that we can't think of that diagram in terms of the explanation given right under it: 


#14
Feb1311, 05:52 PM

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#15
Feb1311, 08:55 PM

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Note the difference  when you start to talk about particles themselves curving spacetime, the picture becomes a lot more complicated, because you have to take into account all sorts of backreactions as the massive particles move. When you think about particles of low enough mass that they have an insignificant effect on curvature, used to probe the structure of a static spacetime around some simple massive object (like a black hole), it becomes a lot easier situation to analyze. The other point you are most likely missing is that it's not space that's curved, it's spacetime. Take a snapshot of the particle relative to the large mass at some "instant in time", it has a certain distance and a certain velocity. Take another snapshot of the particle relative to the large mass at some later "instant in time". Now it is both closer to the large mass, and has a velocity that points towards the large mass. The conclusion is that the mass has somehow "attracted" the particle, even though it's simply been following a geodesic. The notion of what particular surface, i.e. set of points, of spacetime, that corresponds to an "instant in time" is observer dependent  the easiest solution is to use a very slowly moving particle (perhaps one that is even initially not moving at all), in which case the notion of time experienced by said test particle is the same as the notion of time defined by the geometry of the large mass. If you use a short time interval, you'll see that the distance moved by an initally stationary particle is quadratic in time, 1/2 a t^2, so to first order in time it's zero, while the acquired velocity is linear, a t, and that's what you expect the geometry to reproduce. The sort of diagrams you really need to draw to find the velocity cab be found in Donald Marolf, "Spacetime Embedding Diagrams for Black Holes", http://arxiv.org/abs/grqc/9806123. The diagarams that show spatial curvature are interesting, but won't help you understand the answer to your question. However, you'll need to be familiar with the spacetime diagrams of special realtivity and the Lorentz transform to really understand Marolf's particular embedding fully, and the article itself is written from a rather technical viewpoint. 


#16
Feb1411, 10:03 AM

P: 5,632

DrGreg:
The explanation you provided me in the thread I referenced above gave me another insight into how I am still not "visualizing" spacetime effects as accurately as I would like. It's confusing, I suspect for Jorjy and many of us, because we are used to acceleration changing speed,not time and yet in Einstein's spacetime acceleration acts analogous to rotational motion in space (where aceleration is perpendicular to velocity), speed remains constant while direction changes, but acceleration in spacetime also changes passage through TIME...the faster one moves the slower the passage of time for them.... The best way so far for me to "visualize" that is that linear acceleration in space appears as a curve in spacetime..... 


#17
Feb1411, 12:22 PM

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pervect posts



#18
Feb1411, 02:33 PM

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The relative 3velocity of a particle relative to an observer corresponds to the angle between the particle's worldline and the observer's worldline. In curved spacetime this can be defined unambiguously only where the two worldlines intersect. 4acceleration is therefore a change of angle, i.e. the curvature of a worldline, i.e. the reciprocal of the radius of curvature of a worldline. 


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