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I am sorry about the probably naiive nature of this question (I am a software eng, not a physics student):

I (think I) understand the popular curved "trampoline" model of 2-dimensional space, curved by a mass in the center ("bottom") of the trampoline. A free-falling particle will orbit the center, and if the velocity of the particle is larger, the orbit will be farther from the center. This somehow follows from the particle following the "straight" line, where the straight line is defined, that locally, if you take two points on the line, the line is the shortest path between these two points. So if a particle on a larger orbit, suddenly performs "an engine burn" in the direction opposite of traveling, to lose some speed, it will then fall to a lower orbit.

But why will it fall? The velocity vector has the same direction as before, only it is shorter - but the straight line is the same as before, so it should travel along the same (larger) orbit, only slower...

What am I confusing here?

Qmechanic
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Mark Galeck
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1 Answers1

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Because of time.

In 4D spacetime the velocity vector has components in the usual three spatial dimensions and an extra component in time. Even if an object is at rest in space, it is still moving through time.

It's easier to think about things like this if you use a 2D (1D-space + time) analogue. When the object changes its 1D spatial speed, the direction of the overall 2D vector changes. Notably, the overall magnitude of the spacetime velocity is always the same and equal to the speed of light (modulo a $+/-$ sign that depends on an arbitrary convention) $$\vec{v}\cdot\vec{v} = c^2.$$

When an object moves along a geodesic, it doesn't have a destination in mind. It doesn't pick two points and connect them with a minimal path. It looks at the local spacetime and takes a step parallel to its full spacetime velocity vector. If two objects start at the same point but have different spatial speeds (even if they are in the same spatial direction to start), their full velocity vectors are not parallel. The two objects take their steps in different spacetime directions.

Paul T.
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  • Ah yes, the space itself is not curved, it is flat as far as our measurements can tell, it's only the space-time that is curved. I have to study this more :) – Mark Galeck Feb 14 '24 at 18:55
  • That's not quite right either. Both the space and the time are curved. For example the spacetime metric for a point mass (Schwarzschild metric) has curvature in the radial and time directions ($\hat{r}$ and $\hat{t}$ in spherical coordinates). – Paul T. Feb 14 '24 at 18:57
  • In everyday experience the spatial curvature is negligible, though, mainly because velocity of objects we encounter is almost all through time. For relativistic objects like light beams it does matter though (e.g. the extra deflection of light near the Sun in GR as opposed to Newtonian gravity). – Eric Smith Feb 15 '24 at 02:44