How much of gravity is caused by time dilation?

Question

I just watched this video where it is explained how time dilation causes gravity. It is said in this video that time dilation (caused by masses such as the earth) is the only cause for gravity.

But then, what about the curving of space? Does this not contribute to gravity (by objects following the straightest line as depicted in this image)?

Is gravity only caused by gravitational time dilation, if yes, why does the curving of space not have any effect and if no, how much of gravity is caused by time dilation compared to the curving of space?

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Note: I am aware that one cannot actually split the curving of spacetime into two things (the curving of time, and space seperately). I could also ask my question as:

How much of the attractive "force" (well, it is not actually a force) felt by an object in the gravitational field of another object like a planet or star is caused by clocks closer to the object runnning slower (compared to clocks further away) compared to the object moving toward the heavier object due to curved geodesics?

Answer 1

No, it isn't only time curvature that can cause gravitational effects, but the temporal curvature terms (i.e. ones that include time and space, as opposed to just spatial directions) are the only ones we typically notice in everyday life. The Earth curves space and time together in comparable amounts (see this for the exact values) and in theory all of these curvature terms do have effects on trajectories of objects/light, but some we notice and some we don't.

Most everyday objects on Earth (humans, falling apples, etc.) have a much larger temporal component of velocity than spatial component of velocity $\dfrac{d(ct)}{d\tau}>>\dfrac{dx}{d\tau}$. So when finding a geodesic trajectory, the spatial only curvature terms do not significantly contribute.

Basically, you only notice curvature if you move significantly in a curved direction, even if all directions are curved equally. For example, if you walked all the way around the Earth along the equator you could measure that your distance walked is many millions of metersticks different from $2\pi R=2\pi$(distance walked from North Pole to Equator) and therefore you would notice curvature along latitudes. But if you only walked 10 meters North from the equator, it would be difficult to measure whether your arc length deviated from that of a flat space or not, so you would not notice any curvature along longitudes (even though the Earth's surface has equal curvature in all directions). Such is the case in General Relativity for objects that move fast through time, and slow through space. e.g. $\dfrac{d(ct)}{d\tau}>>\dfrac{dx}{d\tau}$

Answer 2

We perceive gravity as a fictitious force as we are pushed off our geodesic by the surface of the Earth, according to the geodesic equation (the version that uses time):

$$ \ddot x^{\mu} = -\Gamma^{\mu}_{\alpha\beta} \dot x^{\alpha}\dot x^{\beta} + -\Gamma^0_{\alpha\beta} \dot x^{\alpha}\dot x^{\beta}\dot x^{\mu}$$

The point of this equation, here, is to show that it's not just time curvature that matters. Alpha and beta run over all the indices.

However, when you are on the Earth's surface sitting at rest:

$$ \dot x^{\mu} \approx (c, 0, 0, 0) $$

and $\alpha$ (or $\beta$) $= 1,2,3$ just don't get an opportunity to enter the calculation.

Also, when you plug in Christoffel symbols for the Schwarzschild metric at $r\gg r_s$, you get the nice result that:

$$ \ddot x^r = -\frac{GM}{(x^r)^2} $$

which is Newton's law of gravitation.

If you get closer to the Schwarzchild radius, move at relativistic velocities, or give the gravitating body lots of angular momentum, then you need more terms than $\Gamma^r_{tt}$ to quantify gravitation.

Back in the weak field of Earth, you can imagine time curvature by considering that path dependence of time as follows:

Two researchers (A & B) in the basement at NIST have atomic clocks, and agree to meet in the top floor conference room in 1 year. A takes the elevator up and waits in the conference room, B sits in the basement for a year, and only then takes the elevator up top. He sees A who then says, "You're late"....their two right angle paths do not meet at the same point in spacetime.

Answer 3

How much of the attractive "force" (well, it is not actually a force) felt by an object in the gravitational field of another object like a planet or star is caused by clocks closer to the object runnning slower (compared to clocks further away) compared to the object moving toward the heavier object due to curved geodesics?

The geodesics are paths in the 4 dimensional spacetime, not in 3-D space only. They include time effects.

The video tries to explain the accelerated movement associated to gravity to differences in the clock rate at different point of an object. It is not the case.

Suppose an object thrown upward. Its velocity will decrease gradually until stops, and then changes direction, and gradually increases until it has the same initial speed, (but opposite direction).

Compare that movement with a plane flying from Paris to Seattle (both cities with similar latitudes about $48^\circ$. Looking at a globe, it is clear that the shortest path (that is part of a great circle, called geodesic) requires keeping a north component of the velocity, that gradually becomes smaller until reach a maximum latitude (about $64^\circ$). Then a small south component appears, that are getting greater until reach Seattle.

If we compare with the previous example, taking latitude as height and longitude as time, the situation is similar to the object thrown upward.

For the plane, the direction of the velocity changes during all the trip due to the curvilinear coordinates. So the movement could be called accelerated. But we know that it is as straight as possible.

For the object, the path is also as straight as possible (a geodesic), but it is shown to us as accelerated, due to the curvilinear coordinates.

Answer 4

The video is just nonsense. Early on, he starts talking about "curvature in time" and saying "time can curve." He just has no idea what he's talking about. Curvature is curvature of spacetime, not time or space separately.

Gravitational time dilation isn't generically even something that can be defined in general relativity. It can only be defined for certain kinds of spacetime that are called static. What is true is that in such a spacetime, the time dilation is closely related to the gravitational potential. And even in this special case, it's not a question of "causing" gravity.

Do yourself a favor and stop trying to learn about science from videos. Read a book.

Answer 5

It's always spacetime, Einstein's ether, which curves. It means that in different states of motion (in respect of gravitating body) you see curvature mostly either in your time or in your space. At the limit of speed of causality c the curvature is seen mostly in space but at the limit of speed zero it's seen in time.

For light ray in vacuum it's curving two times as much as Newtonian prediction because of the lightlike spatial curvature for it itself and the timelike temporal based coordinate curvature for dominant energy density sink of gravitational field. At speed zero at dominant center, the motion curving limits zero (Newtonian model is based on pointlike oriented mass scalar quantities) taking account on energy density gradient.

According GR the curving motion in chosen coordinates is function of Newtonian effect with coefficient 0...2.

This answer may consist of shadows of interpretations, but I believe they can disappear with subtle forgiveness and study.

Here you can find detailed information.

It is highly recommended to consider that there is no crystal clear cause-and-effect sequence with momentary unchanging gravitational position; preferably, we would better say that the gravitational phenomenon is correlated with the curvature of spacetime. We can say about the causal order: curvature and gravitational bodies both result from the matter energy accumulation process.