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?
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?
Best Answer
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. $\dfrac{d(ct)}{d\tau}>>\dfrac{dx}{d\tau}$