Imagine you are driving in a car coming up parallel to the sun on your left. Time moves a bit faster for you on the left side than the right side. This slight speedup makes your left side traverse space faster than the right side, which causes a slight drift to the left (and also makes you spin).
How does this cause a point particle to accelerate towards the sun? Must be something about the gradient, but how does the gradient of time cause you to curve towards the sun?
That's a great question. The answer is, the stuff you are reading in this thread is not right (you figured it out). The real version of the story is, there is this thing called the "Christoffel symbol," which tells you where, at every point in space, you would end up if you went in a certain direction, including which way you would be facing if you went that way. It relates three vectors: your direction of motion, the direction you are currently facing, and the delta to your direction of facing that would result from taking that direction of motion.
If you let your current momentum be your direction of facing, and let the same momentum also specify your direction of motion, the Christoffel symbol tells you what your momentum vector would be after an infinitesimal amount of motion. This can be integrated to find the version of a straight line appropriate for a curved surface (imagine an ant walking straight forwards on the surface of a cone or something), a geodesic. A changing momentum is like a force is acting, so that's gravity.
There is more to learn than that, of course. Many many many books have been written about general relativity and you can read them.
That's a very cool analogy but I might not be understanding something here. Why then do objects that have no light have gravity? If 99% comes from time dilation, why am I stuck to the earth rather than drifting toward light sources?
The point is that mass bends space-time. The amount of bending is dependent on the size of the mass and on the distance from the mass. Even though the Sun is incomparably heavier than the Earth, it is also MUCH farther away from you. So, space-time around the Earth is curved much more towards the center of the Earth than it is towards the center of the Sun. In the mattress analogy, consider a large mattress, with a bowling ball and a car sitting on it. The car will obviously bend the mattress much more, but if you're close to the bowling ball, you'll still fall towards the bowling ball first before both you and it fall towards the car.
So, say you're in an airplane moving directly forward, with the Sun just overhead (and the Earth obviously just below you). The Earth curves spacetime towards it a lot in this area, while the Sun curves it towards itself just a little bit. The overall curvature is such that time still moves more for the bottom of the plane (closer to the Earth) than the top of the plane (closer to the Sun). So, the bottom side moves a little slower than the top side, but the structural integrity of the plane pulls the top side towards the bottom, causing a slight motion towards the Earth - gravity [note that the GP's explanation got the signs a little wrong - time flows slower, not faster, closer to a big mass]. Conversely, if the Earth disappears from the picture and only the Sun remains, now the top part of the plane will move slightly slower, pulling the bottom part towards it, and thus towards the Sun.
If a particle was dropped into the sun’s gravity (not with “horizontal” motion that might cause it to orbit), is it time dilation that causes it to accelerate toward the sun somehow?
I’m going to ask the obvious next question… so if the sun and me in the car are next to each other but stationary, where is the attraction coming from now? As in, time may make the closer side slower, because we’re stationary, there’s no drift etc
You always have to define stationary when it comes to relativity.
There is no way to have a “zero speed orbit”. You’d be on a trajectory straight in to the middle of the sun or away from it (under your own power). The only way to stop is to push away with equal constant acceleration (which looks like “force”). This is what rockets do.
If the sun is on my left, doesn’t that mean time moves a bit slower on my left and the slowdown on the left means I’ve traveled less on my left side? Thus I turn left toward the sun.
It means clocks tick at different rates depending on where they are.
Imagine spacetime as a field of local clocks. Far from the Sun, clocks tick faster. Near the Sun, clocks tick slower. A freely moving object tries to follow the straightest possible path through spacetime. But because the “time axis” changes from place to place, what counts as “straight ahead into the future” tilts slightly inward near the Sun. So the Earth’s path through spacetime curves toward the Sun.
Earth’s spatial speed around the Sun is about 30 km/s. But through spacetime, its “timeward” motion is basically c, 300,000 km/s. So even a tiny tilt in the time direction creates a significant spatial acceleration. That is why the time-warping term dominates for slow massive bodies.
That's right -- the atmosphere stays attached to the Earth mostly thanks to gravity, and the Earth's gravity in GR is almost entirely the gradient in clock rate near the Earth.
Near Earth’s surface, clocks lower down tick very slightly slower than clocks higher up. The change in tick rate is on the order of 10^(-16) per meter. While extremely small, that's enough to generate the familiar 9.8 m/s^2 spatial acceleration we experience. Such a small gradient in clock rates generates macrosopically noticeable spatial accelerations because the "translation" factor is c^2, a tremendously large number.
Now, if I wanted to cover all my bases here, I'd need to point out that gravity does also bend space -- that is just not a relevant factor for "ordinary" gravity acting on relatively slow moving matter (like the Earth itself, or the Earth's atmosphere). For instance, for light itself, spatial bending is just as important (in fact, the gravitational deflection of light by a weak static gravitational field is controlled by a near 50/50 split between spatial and temporal effects). Near a massive black hole, it's not that simple and can't meaningfully be understood in terms of "time" and "space" effects being independently separated.
I’ve seen this described before in terms of how GPS is able to do what it does, so not surprising. But still haven’t explained why time would so dominate the space dimension.
Edit: The response below is dead for some reason, please vouch.
Everything we experience is far larger in time than space, so of course time effects dominate on scales we perceive.
But this just raises the question of what it means to be larger in time than space. You can look at it in terms of multiples of Planck distance or time, but I think there's a more enlightening way to look at it. If you express the speed of light in those Planck units, it's 1. But the speed of light is also the maximum speed of causality. Any causally-bound system must run long enough for chains of causation to propagate, usually far below the speed of light in practice. This means that basically anything that exists within the bounds of our manipulation must be happening at scales where there is far more time involved than space.
We all exist below the diagonal because the diagonal is the bound at which the ways chemistry and biology work no longer even are theoretically possible.
All objects move through spacetime at the speed of light, but a stationary object is moving in the time direction. (And the time dimension has opposite sign to spatial dimensions, so (Lorentzian) rotation's effect on length works opposite to what you'd expect from Euclidean rotations: https://commons.wikimedia.org/wiki/File:Spacetime_diagram_of....) Suppose we drop a test mass from the top of the leaning tower of Pisa. The "forwards through time" direction takes the object deeper into the local gravity well: as far as the test mass is concerned, it's just moving forwards through time according to Newton's First Law, and everything else is accelerating towards it for no apparent reason.
It may help your intuition to consider the extreme case of a black hole. The event horizon is where time is so warped that no possible future trajectories lead outside of the black hole, and you need a magical time machine to escape. (Of course, the best way to gain intuition is to work through the mathematics, either symbolically or with diagrams, rather than reading English-language descriptions.)
There is a sense in which an orbit is a straight line. Obviously, an orbit is not a straight line through space (unless you count the perfect and unobtainable orbit of a beam of light around a black hole, some distance from the event horizon), but we often think of them as spirals through spacetime: there's an argument that really we should think of them as straight lines through spacetime, much like how a great circle is a straight line along the earth's surface.
fortunately that video is more gentle but the math in that youtube channel absolutely melts my brain some days, I can keep up for the first minute but then all bets are off as he dives in and I realize there are some insanely brilliant people out there
Imagine you are driving in a car coming up parallel to the sun on your left. Time moves a bit faster for you on the left side than the right side. This slight speedup makes your left side traverse space faster than the right side, which causes a slight drift to the left (and also makes you spin).
Now just add massive scale and distances.
How does this cause a point particle to accelerate towards the sun? Must be something about the gradient, but how does the gradient of time cause you to curve towards the sun?
That's a great question. The answer is, the stuff you are reading in this thread is not right (you figured it out). The real version of the story is, there is this thing called the "Christoffel symbol," which tells you where, at every point in space, you would end up if you went in a certain direction, including which way you would be facing if you went that way. It relates three vectors: your direction of motion, the direction you are currently facing, and the delta to your direction of facing that would result from taking that direction of motion.
If you let your current momentum be your direction of facing, and let the same momentum also specify your direction of motion, the Christoffel symbol tells you what your momentum vector would be after an infinitesimal amount of motion. This can be integrated to find the version of a straight line appropriate for a curved surface (imagine an ant walking straight forwards on the surface of a cone or something), a geodesic. A changing momentum is like a force is acting, so that's gravity.
There is more to learn than that, of course. Many many many books have been written about general relativity and you can read them.
With QM there’s no pure point particles.
A point particle? You mean that useful mathematical approximation for excitations in a field?
That's a very cool analogy but I might not be understanding something here. Why then do objects that have no light have gravity? If 99% comes from time dilation, why am I stuck to the earth rather than drifting toward light sources?
Light has nothing to do with this.
The point is that mass bends space-time. The amount of bending is dependent on the size of the mass and on the distance from the mass. Even though the Sun is incomparably heavier than the Earth, it is also MUCH farther away from you. So, space-time around the Earth is curved much more towards the center of the Earth than it is towards the center of the Sun. In the mattress analogy, consider a large mattress, with a bowling ball and a car sitting on it. The car will obviously bend the mattress much more, but if you're close to the bowling ball, you'll still fall towards the bowling ball first before both you and it fall towards the car.
So, say you're in an airplane moving directly forward, with the Sun just overhead (and the Earth obviously just below you). The Earth curves spacetime towards it a lot in this area, while the Sun curves it towards itself just a little bit. The overall curvature is such that time still moves more for the bottom of the plane (closer to the Earth) than the top of the plane (closer to the Sun). So, the bottom side moves a little slower than the top side, but the structural integrity of the plane pulls the top side towards the bottom, causing a slight motion towards the Earth - gravity [note that the GP's explanation got the signs a little wrong - time flows slower, not faster, closer to a big mass]. Conversely, if the Earth disappears from the picture and only the Sun remains, now the top part of the plane will move slightly slower, pulling the bottom part towards it, and thus towards the Sun.
One nit: Time moves a bit slower on the sun's side.
Other than that, thank you for a very clear explanation.
If a particle was dropped into the sun’s gravity (not with “horizontal” motion that might cause it to orbit), is it time dilation that causes it to accelerate toward the sun somehow?
I’m going to ask the obvious next question… so if the sun and me in the car are next to each other but stationary, where is the attraction coming from now? As in, time may make the closer side slower, because we’re stationary, there’s no drift etc
You always have to define stationary when it comes to relativity.
There is no way to have a “zero speed orbit”. You’d be on a trajectory straight in to the middle of the sun or away from it (under your own power). The only way to stop is to push away with equal constant acceleration (which looks like “force”). This is what rockets do.
If the sun is on my left, doesn’t that mean time moves a bit slower on my left and the slowdown on the left means I’ve traveled less on my left side? Thus I turn left toward the sun.
I believe that is correct.
This was a helpful visualization: https://youtu.be/U_sI9agWmEw?si=MItDfnTx1-oT_qX_
Also very well explained by PBS Spacetime here https://youtu.be/UKxQTvqcpSg
I skimmed the video, but it's definitely the same example that finally made this make sense to me.
I always hated the ball and sheet example simply because it was describing gravity with gravity. It felt fundamentally wrong.
It changes the geodesic that Earth follows, from a straight line (in 3D space) to a curved one (an ellipse).
And also, how does one bend time?
It means clocks tick at different rates depending on where they are.
Imagine spacetime as a field of local clocks. Far from the Sun, clocks tick faster. Near the Sun, clocks tick slower. A freely moving object tries to follow the straightest possible path through spacetime. But because the “time axis” changes from place to place, what counts as “straight ahead into the future” tilts slightly inward near the Sun. So the Earth’s path through spacetime curves toward the Sun.
Earth’s spatial speed around the Sun is about 30 km/s. But through spacetime, its “timeward” motion is basically c, 300,000 km/s. So even a tiny tilt in the time direction creates a significant spatial acceleration. That is why the time-warping term dominates for slow massive bodies.
Does the atmosphere stay attached to the earth due to the bending of time?
That's right -- the atmosphere stays attached to the Earth mostly thanks to gravity, and the Earth's gravity in GR is almost entirely the gradient in clock rate near the Earth.
Near Earth’s surface, clocks lower down tick very slightly slower than clocks higher up. The change in tick rate is on the order of 10^(-16) per meter. While extremely small, that's enough to generate the familiar 9.8 m/s^2 spatial acceleration we experience. Such a small gradient in clock rates generates macrosopically noticeable spatial accelerations because the "translation" factor is c^2, a tremendously large number.
Now, if I wanted to cover all my bases here, I'd need to point out that gravity does also bend space -- that is just not a relevant factor for "ordinary" gravity acting on relatively slow moving matter (like the Earth itself, or the Earth's atmosphere). For instance, for light itself, spatial bending is just as important (in fact, the gravitational deflection of light by a weak static gravitational field is controlled by a near 50/50 split between spatial and temporal effects). Near a massive black hole, it's not that simple and can't meaningfully be understood in terms of "time" and "space" effects being independently separated.
I’ve seen this described before in terms of how GPS is able to do what it does, so not surprising. But still haven’t explained why time would so dominate the space dimension.
Edit: The response below is dead for some reason, please vouch.
Everything we experience is far larger in time than space, so of course time effects dominate on scales we perceive.
But this just raises the question of what it means to be larger in time than space. You can look at it in terms of multiples of Planck distance or time, but I think there's a more enlightening way to look at it. If you express the speed of light in those Planck units, it's 1. But the speed of light is also the maximum speed of causality. Any causally-bound system must run long enough for chains of causation to propagate, usually far below the speed of light in practice. This means that basically anything that exists within the bounds of our manipulation must be happening at scales where there is far more time involved than space.
We all exist below the diagonal because the diagonal is the bound at which the ways chemistry and biology work no longer even are theoretically possible.
[dead]
It's much simpler than anyone thought:
https://youtu.be/A-2XQQDD6QQ
Same way you bend space. In GR, time is just another dimension of a slightly different flavor.
All objects move through spacetime at the speed of light, but a stationary object is moving in the time direction. (And the time dimension has opposite sign to spatial dimensions, so (Lorentzian) rotation's effect on length works opposite to what you'd expect from Euclidean rotations: https://commons.wikimedia.org/wiki/File:Spacetime_diagram_of....) Suppose we drop a test mass from the top of the leaning tower of Pisa. The "forwards through time" direction takes the object deeper into the local gravity well: as far as the test mass is concerned, it's just moving forwards through time according to Newton's First Law, and everything else is accelerating towards it for no apparent reason.
It may help your intuition to consider the extreme case of a black hole. The event horizon is where time is so warped that no possible future trajectories lead outside of the black hole, and you need a magical time machine to escape. (Of course, the best way to gain intuition is to work through the mathematics, either symbolically or with diagrams, rather than reading English-language descriptions.)
There is a sense in which an orbit is a straight line. Obviously, an orbit is not a straight line through space (unless you count the perfect and unobtainable orbit of a beam of light around a black hole, some distance from the event horizon), but we often think of them as spirals through spacetime: there's an argument that really we should think of them as straight lines through spacetime, much like how a great circle is a straight line along the earth's surface.
it's the speed of causality (limit of information transfer)
https://www.youtube.com/watch?v=8yhk1EZq9tY
fortunately that video is more gentle but the math in that youtube channel absolutely melts my brain some days, I can keep up for the first minute but then all bets are off as he dives in and I realize there are some insanely brilliant people out there