# What is the relationship between the flow rate of time and gravitational field?

I have heard, qualitatively, about the fact that time's rate of flow slows when the gravitational field is "strong". Here I am looking for some more rigorous descriptions of this phenomenon.

My first question is, suppose there are two points--A and B, A is ahead of B in the impending falling--in a uniform gravitational field with a field strength $g$, how would the rate of flow in time differ among point A and point B which are distance $d$ apart in the direction of the field? (How would the rate of flow of time even be expressed?)

Second question, suppose there are two point sized clocks initially at point A and B, both at rest initially with time readings of TA and TB which are both reading zero, what would the readings of TB be, as seen by clock-A after its own reading goes to TA=t? (In addition, how would things differ if the clocks are fixated in space?)

My third question is, assuming the rate of flow of time can be quantified, what is the expression for χ=f(r) where χ stands for the rate at which time flows and r stands for the distance between a point of interest and a massive object?

First it is important to note that all of the effects you have described underly the rules of general relativity and can not be described in Newtonian mechanics.

1. The time dilation between two observers (around a black hole) is given by:

$\delta t = {\delta t_0}/ \sqrt{1-R_s/r}$

Where $R_s$ is the Schwarzschild radius. Notice that as $r$ reaches the value of the Schwarzschild radius, the time dilation goes to infinity meaning that the time effectively stops. This is the point of the event horizon of a black hole.

1. For this question it is important to know which clock is closer to the black hole because its time will be more dilated than the clock further away. Please Specify and I will give a more complete answer.

2. The rate of flow of time is not absolute but can only be quantified relative to a second observer. Therefore there is no absolute formula for the rate of flow of time.

• I set the question in the context of a uniform gravitational field, but if that's not possible, a black is fine too although I would like to know why is a uniform field not possible for time dilation. I did in fact say in the question that A is nearer to the source of gravity, in the case of the black hole, A is closer to the black hole. – user289661 Feb 24 '16 at 4:02
• Are both A and B in the uniform gravitational field? – Jaywalker Feb 24 '16 at 7:24
• The problem is that a uniform field wont create a geodesic curvature of space time and therefore does not follow general relativity. – Jaywalker Feb 24 '16 at 7:25
• Thank you for pointing that out, I in fact didn't know that attribute of the uniform field, I would appreciate it if you could explain what is meant by "geodesic curvature" if that is reasonable to do in this format. In addition, as opposed to a uniform field, what will happen if the objects are near a black hole instead? – user289661 Feb 25 '16 at 0:31

I have heard, qualitatively, about the fact that time's rate of flow slows when the gravitational field is "strong".

When learning physics an important step is to learn which things used to be equal that aren't anymore. For instance in Newtonian mechanics $m\vec a =\vec F=\mathrm d \vec o/\mathrm d t$ but in relatively $m\vec a \neq \mathrm d \vec o/\mathrm d t$ so they can't both equal the force. And learning that one if them equals the force wouldn't help you if you didn't learn they aren't equal.

Why did I bring that up. Becasue one thing that changed in relativity is there is no objective way to say how far apart things are (rulers moving differently have their parts take take different paths through 4d spacetime and have their markings line up different ways). And there is no objective way to say the time between two events (clocks moving differently between the two events lay their ticks down at different events so have a different number of ticks between the two events).

You can still make predictions about what clocks and rulers do, but you have to specify these details.

My first question is, suppose there are two points--A and B,

Already, when you say points you imagine two points at the same time, the idea of same -time isn't actually a thing, just like $m\vec a \neq \mathrm d \vec o/\mathrm d t.$

You could try to specify a family of observers, each of which fires thrusters so they stay a fixed unchanging distance from each other as they stay above a star or planet. You could assign distances between them and since they don't change in time you could even assign coordinates to them and call them places and points and such. But this was one choice of many.

And it's a nice choice that these distances don't change in time. And they are nice that they seem familiar to the everyday places and points we draw on maps (we don't use rockets, we use the earth compressing under our feet and buildings providing a pressure underneath that keeps us from going towards the center of the earth, but it doesn't matter how that was prevented, it happens).

But it's bad in the sense that the coordinates are not an inertial frame. And it's tricky in that clocks carried by these people at these fixed coordinates actually tick at different rates. So it's tricky to say when something is happening.

On the earth the clocks tick at rates that are numerically close so you don't even notice unless you have a super accurate clock (the difference are smaller than the inaccuracies of most cheap clocks). So it's like when your distances are much smaller than the markings on your ruler, you can't really notice it easily.

A is ahead of B in the impending falling--in a uniform gravitational field with a field strength ,

A uniform field isn't going to have things change much. The way you get a uniform field is to be far from the center and to have your distance to the center not change much percentage wise. So percentage wise your field didn't change much. But then nothing is changing very much. So you basically said don't let them fall very much and don't let then be far from each other (since either of those would require the field to change a noticable amount).

how would the rate of flow in time differ among point A and point B which are distance  apart in the direction of the field?

And now you further insist they stay a fixed distance apart. Even if you fix that coordinate system of the people with rockets staying fixed distances apart, falling things don't stay a fixed distance apart.

(How would the rate of flow of time even be expressed?)

The clocks deeper towards the center see ticks arrive from the clocks up above at a faster rate than they see their own ticks. And the clocks farther up from the center see ticks arrive from the clocks down below at a slower rate than they see their own ticks. And this happens even if they ship their clocks down and up to each other or manufacture new ones using the same instructions and materials. Clock tick based on the metric along the path in 4d the clocks travel. That's what clocks do. And be metric is different deeper in the well.

It's not totally inappropriate to imagine there is a time, but it is inappropriate to imagine that clocks measure it. Clocks measure the metric along the path in 4d that the clock travels.

Second question

It's supposed to be one question per post.

suppose there are two point sized clocks initially at point A and B, both at rest initially with time readings of TA and TB which are both reading zero, what would the readings of TB be, as seen by clock-A after its own reading goes to TA=t?

Whenever you see a clock reading from a distant clock you see it from the past. A clock an arms length away is from 1ns in the past. If you see a clock on Mars, the image you see now is from 10 minutes ago. Light takes time to get to you. But if you have those locations with the rocket thrusters then the distances don't change. So you saw a reading of $-1ns$ when your own clock was $0s$ and then $1ns$ later ($1ns$ later to you) your own clock reads $1ns$ but the clock farther from the center sends a signal that arrives the same time that is still a negative reading (not $0ns$) and you can compute that difference by comparing how often your clock ticks to how often their clock tick.

One reference it to have everyone compare their clocks rate to a clock that is a few light years away. Then yours ticks at a rate of $\sqrt{1-r_s/r}$ compared to that distant clock. So if you are at $r=r_1$ and the other clock uo higher is at a larger $r=r_2>r_1$ then the ratio of rates is $\sqrt{1-r_s/r_1}/\sqrt{1-r_s/r_2}$ and as a warning the $r$ is not the distance to the center of the planet. And $r_2-r_1$ isn't the difference in height. Instead, when you are a certain distance $d$ from the center there is a spherical shell of area $4\pi r^2$ and so the distance $d$ gives you the coordinate $r$ and a flat space might make you think $d$ and $r$ are the same, but they are not. Just like on a funnel you could take circles of circumference $2\pi r$ and get an $r$ from that but going from $r_2$ to $r_1$ would require travelling more than $r_2-r_1$ because that distance would just move you from one circle to another circle in the same plane but moving on a funnel requires also going down. Similarly the distance between the two heights is larger than $r_2-r_1$ we named the coordinate to give a surface area of $4\pi r^2$ we could have picked a coordinate that gives distances between the layers easily and the area would have been complicated, or we could pick a coordinate that gives the surface area if each layer in a simple way $A=4\pi r^2$ and gives distances between layers in a complicated way. The $r$ in the time dilation formula is the areal (area-l) coordinate.

(In addition, how would things differ if the clocks are fixated in space?)

This doesn't have a fixed meaning. If they are fixed relative to those people on thrusters then you get what I said before. If they are freely falling they won't stay a fixed distance apart (uniform gravitational fields aren't possible) and so that lag time for what you see is changing so the $1ns$ gap will be changing.

But you are asking all the wrong questions. The clocks don't measure time, they measure the metric along a curve. So you have to find the metric for the actual situation, find the paths of the clocks you care about. Compute the metric along that path. Then you know how much that clock ticked along that path in that metric. It all matters. And there isn't something else going on.

My third question is, assuming the rate of flow of time can be quantified, what is the expression for χ=f(r) where χ stands for the rate at which time flows and r stands for the distance between a point of interest and a massive object?

The distance between a massive objects center and where you are isn't s thing. In particular if your planet is collapsing (even just collapsing slowly) then even when you stay at a constant areal $r$ (stay on a spherical surface of surface area $4\pi r^2$) the distance to the center of he object changes.

Let's go back to the funnel. Put a quarter (large coin) in the funnel. The quarter is the inside of your planet. The funnel is the curved spacetime outside your planet. The area $r$ is related to the circumference of the circle where you are. Smaller circumferences, closer to the center. If you measure the distance to the center, distances are in the spacetime (that's where rulers are) so they go along the funnel until the get to the quarter then they go along the quarter until they get to the center. So there is a fixed distance for each circumference. It's just that when you circumference goes from $2\pi r_2$ to $2\pi r_1$ you traveled more than $r_2-r_1$ (unless both parts were so close as to be inside).

Now your planet gets smaller. Replace the quarter (large coin) with a dime (small coin). The circumference outside didn't change. But now you get those larger distances all the way to the dime so the center has move farther away.

That really happens. You are staying on a spherical shell of area $4\pi r_1^2$ and your friend is up higher staying on a spherical shell of area $4\pi r_2^2$ and neither if you "moved" (sure you need your thrusters to stay on that shell but you aren't moving off your shell) and neither did anyone on any location on either shell. But if the planet collapsed a little bit you are now all "farther away" from the center.

But none of your clocks changed how they tick. And they didn't change how they ticked compared to each other. And they didn't change how they ticked compared to a spherical shell that was super large. So using areal $r$ was a good choice. And using distance from the center of the object is a truly objectively terrible way to quantify it.

And you really need to unlearn the stuff that doesn't hold in general relativity or every thought and question will be a disaster. You can't learn relativity even a little bit while still thinking wrong things hold.