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I understand that all the particles move in spacetime at $c$. A massive particle affects the spacetime, rendering it bent or curved.

In the absence of a massive particle, the said particle (in unbent spacetime) will have a zero spatial and a non-zero temporal relative speed. In the presence of a massive particle, the particle (in bent spacetime) will have a non zero spatial and temporal relative speed, such that the speed in spacetime remains $c$.

I can digest the fact that a particle will always move forward in time w.r.t. to the massive particle, accepting it as law of nature, I don't quite understand why does a particle (in the presence of a massive particle) move in space? Could it be because the spacetime itself moves? If there is no explanation, I can accept that as law of nature as well. (But I had to ask first :)

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It might help to think of an object having four possible directions of motion: $\hat x, \hat y, \hat z,$ and $\hat t$. Express time as just another distance unit, with the special quality that every time a clock ticks, you go one unit that way. Get rid of two of the spatial directions (confine the motion of the object to the plane defined by $x$ and $t$). In flat spacetime, from a distant comoving frame, we and our distant friend are both moving in the $\hat t$ direction. Every time our clock ticks, we see them move an equal distance (one clock tick) in the $\hat t$ direction. That is, $dt_{them}/dt_{us} = 1$

If we put our friend close to a gravitational field from which we are distant, with the gradient in the $\hat x$ direction. Which is to say, we put them in a curved spacetime while we are still in our flat spacetime. Then their local spacetime axis is pointed a little bit different from our local spacetime axis. They keep moving at the same rate (one distance unit per tick of our clock) and in the same direction (their $\hat t$), but their local universe is at an angle to our local universe. The next time our clock ticks, from our perspective, we see them cover a tiny bit less than one tick worth of our $\hat t$ distance (their clock doesn't quite tick yet), and they've also covered a little bit of our $\hat x$ distance. Now we're no longer stationary with respect to each other - they've moved away from us $\Delta x$ in time $\Delta t$, and that's a nonzero relative velocity, even though we're still stationary in our frame and they're still stationary in their frame.

So, the next time our clock ticks, we see them move away from us at their velocity, but their local spacetime is still at an angle to ours, so we see their velocity increase again by the same amount. They're not just moving away from us, they're accelerating away from us... even though we're stationary in our frame and they're stationary in their frame.

So, we have:

$dt_{them}/dt_{us} < 1$: there is time dilation

$d^2x/dt_{us}^2 > 0$: we see them accelerate

$d^2x/dt_{them}^2 = 0$: they have no proper acceleration

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The instantaneous motion for a particle in spacetime will follow a geodesic. This is Newtons first law of motion transplanted to curved spacetime by Einstein. However, there is no reason why it should move at all. This is why physics is an expetimental science and not a purely theoretical construct.

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The easiest way to imagine this is to draw a 2d spacetime diagram for flat space, with the time axis upright and the x axis horizontal. In flat space in the absence of any force an object will move up the t axis vertically, ie without changing its position on the x axis.

Now, if you curve the t axis somewhat, you will see that an object moving up it no longer remains at the same place on the x axis. Simply by following a curved t axis, the object not only moves on the x axis but accelerates along it.

The explanation I have given is very simplified- general relativity is much more complicated- but hopefully that will show you why curving spacetime causes objects to accelerate.

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    $\begingroup$ I think when it comes to General Relativity, we have to treat the word "acceleration" with a bit more nuance. If we're free falling towards the center of the Earth, we're not accelerating since we're following a geodesic. In fact, we have to accelerate in order to stand still and not free fall. Acceleration in the context of GR means to deviate from a geodesic, which is different from moving along geodesics in curved spacetime. $\endgroup$
    – RMC777
    Sep 25 at 6:44
  • $\begingroup$ @RMC777. I think Marco meant acceleration in spatial dimension. So it's a valid argument. If time is dilated, the only option is to move in space with a non zero speed to maintain c in spacetime. The answer to why time is dilated is a different question altogether. Hence my up-vote to Marcos answer $\endgroup$
    – bitsabhi
    Sep 25 at 6:59
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    $\begingroup$ @bitsabhi Many thanks for your comment. I thought the OP was asking a conceptual question about why objects move with increasing speed in curved spacetime, so the use of the word accelerate to describe the increasing speed of the object through space seemed to fit with the plain English tone of the question. $\endgroup$ Sep 25 at 7:31
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What the Einstein Field Equations (EFE) tell you is that the geometry of spacetime $G_{a b}$ is affected by the energy/matter present in said spacetime $T_{a b}$; or as Wheeler aptly put it "Spacetime tells matter how to move, matter tells spacetime how to curve". Said equations are the following

\begin{equation} G_{a b} := R_{a b} - \frac{1}{2} R g_{a b} = \frac{8 \pi G}{c^4} T_{a b} \end{equation}

Where $R_{a b}, \ R, \ g_{a b}$ are the Ricci tensor, the Ricci scalar, and the metric respectively.

Imagine at first you have a test particle so that the curvature it itself causes on the metric is negligent. In this setting, this particle is in Minkowski spacetime, that is a purely flat spacetime. Now, if you put a massive particle next to it (or technically anything described by $T_{a b}$), spacetime will curve as described by the EFE.

This curvature will change the geometry of spacetime, and therefore change the trajectories at which inertial objects move. If our test particle is following a geodesic (that is if it is in inertial motion), when our new massive particle is introduced, the geodesic of our test particle will change accordingly, and therefore "move" towards the massive object. The thing we have to keep in mind is that our test particle isn't really "moving", rather it is free-falling towards our massive object simply due to the curvature induced by it.

As an analogy, consider how meridians on a sphere (which are geodesics) starting from the north pole diverge towards the equator and then converge towards the south pole (where they will meet again) simply due to the sphere's curvature. Nothing's pushing nor pulling the meridians, they're simply straight lines on a sphere that end up intersecting each other due to the sphere's geometry.

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  • $\begingroup$ Thanks for the answer but it doesn't really say why do particles have to move spatially in a curved spacetime in the first place. $\endgroup$
    – bitsabhi
    Sep 25 at 7:03
  • $\begingroup$ @bitsabhi yes it does. It is due to the curvature that the energy-momentum tensor $T_{a b}$ induces on the geometry $G_{a b}$ of your spacetime. Curvature is the only reason why massive things "fall" towards each other. Inertial objects follow geodesics, which in curved spacetime forces them to move through space and not only through time. $\endgroup$
    – RMC777
    Sep 25 at 7:17

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