Looking for Layman's Physical Explanation for Gravitational Time Dilation

[stuper]

I'm looking for a simple physical explanation of gravitational time dilation, so I can provide a brief explanation to interested laypeople (not to mention just to help myself understand better). Does anybody want to take a crack at it? Or am I asking for the impossible?

 

[Naty1]

Try reading the responses to this thread,

https://www.physicsforums.com/showthread.php?t=275018&highlight=time+dilation

then come back and refine your question, if necessary.

 

[George Jones]

Also, for a hand-waving explanation, try reading

https://www.physicsforums.com/showpost.php?p=1320213&postcount=4.

 

[Jonathan Scott]

I'm looking for a simple physical explanation of gravitational time dilation, so I can provide a brief explanation to interested laypeople (not to mention just to help myself understand better). Does anybody want to take a crack at it? Or am I asking for the impossible?

I don't know exactly what you're looking for, but the basic principle is simply that where in Newtonian gravity you have potential energy, in relativistic gravity you have time dilation.

For example, if you move something of mass m from distance r₁ to distance r₂ from a mass M, the change in potential energy is

-m(GM/r₂ - GM/r₁)

= -GMm(1/r₂ - 1/r₁)

Dividing by the rest energy mc², that means the fraction by which the time rate changes is as follows:

-(GM/c²) (1/r₂ - 1/r₁)

This is only an approximation, but a very accurate one for weak gravitational fields such as those within the solar system.

 

[stuper]

Thanks for the responses so far. George, I need to spend a bit more time on your's -- it looks promising, although at first glance it seems to be backwards from what I understood from the book I'm currently reading (Schutz's "Gravity from the ground up").

Here's what I'm looking for more precisely. Let's say I'm having a casual conversation with a relatively (NPI) non-scientific friend, and I mention that time runs slower near a large mass than it does in outer space. He/she is immediately going to ask for an explanation. How am I going to explain this idea in non-technical language without getting into the equivalence principle and the frequency of light waves, etc.? Is there any short, simple way of giving a layperson at least a little inkling of the physical basis behind this phenomenon?

At this point, the simplest explanation I've heard is that both time and space become more curved and thus more dense near a large mass. However, when I think about this it seems to me that time should move faster near a large mass. If you imagine time/space laid out in a grid sort of like latitude/longitude lines, more curved and dense means that the lines are closer together, which means it should be easier to cross more lines (i.e., time moves faster, which I know is not the right conclusion). I'm know I'm thinking about this backwards somehow, but I can't quite see how closer together grid lines translate to slower time.

Then again, maybe somebody else has an even better way of explaining it without the grid lines idea.

 

[Jonathan Scott]

Thanks for the responses so far. George, I need to spend a bit more time on your's -- it looks promising, although at first glance it seems to be backwards from what I understood from the book I'm currently reading (Schutz's "Gravity from the ground up").

Here's what I'm looking for more precisely. Let's say I'm having a casual conversation with a relatively (NPI) non-scientific friend, and I mention that time runs slower near a large mass than it does in outer space. He/she is immediately going to ask for an explanation. How am I going to explain this idea in non-technical language without getting into the equivalence principle and the frequency of light waves, etc.? Is there any short, simple way of giving a layperson at least a little inkling of the physical basis behind this phenomenon?

At this point, the simplest explanation I've heard is that both time and space become more curved and thus more dense near a large mass. However, when I think about this it seems to me that time should move faster near a large mass. If you imagine time/space laid out in a grid sort of like latitude/longitude lines, more curved and dense means that the lines are closer together, which means it should be easier to cross more lines (i.e., time moves faster, which I know is not the right conclusion). I'm know I'm thinking about this backwards somehow, but I can't quite see how closer together grid lines translate to slower time.

Then again, maybe somebody else has an even better way of explaining it without the grid lines idea.

The idea that space-time is "denser" near masses is a possible approach. One way to think about it is to assume that the coordinate speed of light c gets slower closer to masses, by a fraction 2Gm/rc². In an isotropic coordinate system (one where the variation of c is the same in all directions) rulers shrink approximately by Gm/rc² and clocks run slower by the same factor. This means that locally observers can't detect this change in the speed of light.

 

[A.T.]

I'm looking for a simple physical explanation of gravitational time dilation, so I can provide a brief explanation to interested laypeople (not to mention just to help myself understand better).

Not sure what you mean with "physical explanation". Here is a simple explanation how gravitational time dilation follows from the geometrical model of GR.
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

 

[DrGreg]

Consider an apple dropped from the top of a tall building. According to Newtonian theory, the building is stationary and the apple accelerates downwards. According to general relativity, the apple is stationary (or, more accurately, "inertial") and the building accelerates upwards.

The acceleration due to gravity varies with height: it's higher at low altitudes. So the bottom of the building accelerates more than the top of the building. Yet the height of the building is constant. The only way to make sense of this is that the two accelerations are being measured by different clocks ticking at different rates.

That's a gross over-simplification, but in essence that's what gravitational time dilation is. (It's an oversimplification because distance as well as time is affected and the argument above isn't good enough to distinguish both effects.)

 

[A.T.]

The acceleration due to gravity varies with height: it's higher at low altitudes. So the bottom of the building accelerates more than the top of the building. Yet the height of the building is constant. The only way to make sense of this is that the two accelerations are being measured by different clocks ticking at different rates.

Sounds very misleading to me. In a homogeneous G-field the acceleration doesn't vary with height, but clocks rates still do.

 

[DrGreg]

Sounds very misleading to me. In a homogeneous G-field the acceleration doesn't vary with height, but clocks rates still do.

If, by a "homogeneous G-field" you mean a "uniform" field, i.e. as described by the Rindler metric, you are wrong, the acceleration does vary with height. If you mean something else then maybe it doesn't, but I suspect in that case there might be no dilation either.

 

[A.T.]

If, by a "homogeneous G-field" you mean a "uniform" field, i.e. as described by the Rindler metric, you are wrong, the acceleration does vary with height.

What if not acceleration is uniform in a uniform gravitational field?

 

[DrGreg]

What if not acceleration is uniform in a uniform gravitational field?

Sorry, I don't understand your question. Can you rephrase it?

 

[A.T.]

Sorry, I don't understand your question. Can you rephrase it?

I'm just trying to understand, what quantity is "uniform" in a "uniform gravitational field", since you claim that acceleration is different at different places in such a field.

Going back to the original question: I think that gravitational time dilation would occur in a field with the same (non zero) acceleration at every point. By Analogy: In rocket with constant acceleration, the front clock runs faster than the back one, while both experience the same acceleration.

 

[DrGreg]

Going back to the original question: I think that gravitational time dilation would occur in a field with the same (non zero) acceleration at every point. By Analogy: In rocket with constant acceleration, the front clock runs faster than the back one, while both experience the same acceleration.

Check out "Born rigid acceleration". The back of the rocket accelerates more than the front.

To maintain a constant length in its own frame, its length must contract in an inertial frame. To contract, the back must accelerate more that the front.

If both ends had the same acceleration, the rocket would get longer in its own rest frame. See "Bell's spaceship paradox".

I'm just trying to understand, what quantity is "uniform" in a "uniform gravitational field", since you claim that acceleration is different at different places in such a field.

Because of the "Born-rigid" effect, a "uniform gravitational field" is usually considered to be one in which free-falling objects that are initially stationary relative to each other will remain stationary relative to each other, and this implies a change in proper-acceleration over distance.

The Rindler metric applies equally to a Born-rigid accelerating rocket in empty space and a "stationary" observer in a "uniform gravitational field". Both spacetimes have zero curvature but exhibit "gravitational" time dilation.

I think I read somewhere (but I could be wrong) that a gravitational field with constant "acceleration" everywhere is impossible.

 

[A.T.]

Thanks for the explanations DrGreg.

I think I read somewhere (but I could be wrong) that a gravitational field with constant "acceleration" everywhere is impossible.

I didn't know this. But I assumed that in such a (potentially impossible field) there still would be a red/blue shift of light and therefore gravitational time dilation.

 

[feynmann]

Alice and Bob are in a rocket accelerating upward in empty space. Alice, in the nose, emits signals at equal intervals on a clock there. The acceleration means that Bob, in the tail, measures a smaller interval between the received signals, why?
Einstein's equivalence principle says that acceleration and gravity are equivalent. So it should happen in a uniform gravitational field too.

 

[Dale]

I'm looking for a simple physical explanation of gravitational time dilation

Sometimes it helps to describe some experimental result first, and then describe the theory that explains it. So in this case I would start with a brief description of a gravitational redshift experiment, and then relate that to the speed of a clock, and then you have gravitational time dilation.

 

[feynmann]

Kip Thorne's Black Holes and Time Warps. In it, he proposes a thought experiment (which he attributes to Einstein) which demonstrates gravitational time dilation.
--------------------------------------------------------------------

Take 2 identical clocks. Place one on the floor of a room next to a large hole, and attach the other to the room's ceiling by a short string.

The ceiling clock emits pulses of light at each tick and directs them downwards toward the floor clock. Immediately before the first pulse, cut the string so that the ceiling clock is now falling freely. If it is ticking fast enough, then the duration between the first few ticks will be governed by the 'ceiling' time, as it will not have fallen appreciably yet.

Immediately before the first pulse hits the floor clock, drop the floor clock into the hole. Similarly, this clock will feel 'floor' time for the first few ticks.

Now, because the ceiling clock was dropped before the floor clock, its downward speed is always greater than the floor clock. This implies that the floor clock will see the ceiling clock's pulses Doppler shifted (slightly faster). Since the time between pulses was regulated by the ceiling's time flow, this means that time must flow more slowly near the floor than near the ceiling; in other words, gravity must dilate the flow of time.

---------------------------------------------------------------------

 

[yuiop]

... and I mention that time runs slower near a large mass than it does in outer space. He/she is immediately going to ask for an explanation. How am I going to explain this idea in non-technical language without getting into the equivalence principle and the frequency of light waves, etc.? Is there any short, simple way of giving a layperson at least a little inkling of the physical basis behind this phenomenon?

First, an easy to visualise physical illustration of what is happening, before attempting an explanation. Imagine you have a cannon that is connected to clock and is designed to fire one baseball per minute. This device is lowered into a hole that goes deep into the massive body. (Imagine it is a bit more massive and dense than the Earth). Now, when the cannon is fired upwards the baseballs arrive at the surface at intervals of say 61 seconds, according to the clock of the observer at the surface. In this example it is not the frequency of individual baseballs that changing unlike the example given using photons and it easier to see in this case that the clock controlling the cannon really is running slower than the clock at the surface. Now the principle behind this is that whatever must happen to ensure a local observer will always measure the local speed of light as c will happen.

Another example was hinted at in another post and although it uses the equivalence principle, it is fairly intuitive if you accept that clocks moving at relativistic speeds relative to an observer, run slower (as predicted by Special Relativity). Consider a very long rocket that is so long it takes light one second to traverse the length of the rocket. Now imagine a signalling device in the nose of the rocket sends signals to the base of the rocket at one second intervals, as controlled by a clock in the nose of the rocket. Now if the rocket accelerates upwards at a rate of 10 m/s/s, the base of the rocket is always moving 10 m/s faster than the nose of rocket by the time a signal tranverses the length of the rocket. A clock at the base of the rocket will be running slower (due to SR time dilation) at the time it receives the signal, relative to the rate of the clock in the nose the time the signal was emitted. The observer in the base of the rocket perceives the signals to be arriving at interval of less than one second because his clock is effectively running slower at the time he receives the signals relative to the clock at the time the signals were sent. I am igoring length contraction of the rocket in this example, as that is probably getting too complicated for a laypersons explanation to a friend.

Now the equivalence principle says that if the rocket was standing on the surface of a gravitational body with a gravitational acceleration of 10m/s/s the measurements inside the rocket would be exactly the same and in order for that to be so, the lower clock has to run slower due to the presence of the gravitational field.

At this point, the simplest explanation I've heard is that both time and space become more curved and thus more dense near a large mass. However, when I think about this it seems to me that time should move faster near a large mass. If you imagine time/space laid out in a grid sort of like latitude/longitude lines, more curved and dense means that the lines are closer together, which means it should be easier to cross more lines (i.e., time moves faster, which I know is not the right conclusion). I'm know I'm thinking about this backwards somehow, but I can't quite see how closer together grid lines translate to slower time.

Then again, maybe somebody else has an even better way of explaining it without the grid lines idea.

It would be easier for light to cross more lines in the grid where the grid lines are closer to each other and that would make light appear to be moving at greater than c according to a local observer, so everything has to slow down in the denser parts of the grid to make the speed of light constant where ever you are. It is a bit mysterious as to why nature insists that everyone should observe the speed of light to be locally constant, but that is the way it seems to work.

 

[A.T.]

At this point, the simplest explanation I've heard is that both time and space become more curved and thus more dense near a large mass. However, when I think about this it seems to me that time should move faster near a large mass. If you imagine time/space laid out in a grid sort of like latitude/longitude lines, more curved and dense means that the lines are closer together, which means it should be easier to cross more lines (i.e., time moves faster, which I know is not the right conclusion).

You are confused, because you mix up two different(but equivalent) ways to visualize curvature:

1)
A grid with different distances between grid lines, but same density everywhere. Here the distances between grid lines(of space and proper time) are greater near a mass, and every object advances with the same constant velocity(with respect to coordinate time). So objects near the mass need more coordinate time to reach the next grid line of their proper time.

For the laymen: Imagine you track a plane flying with a constant velocity to the west on a globe. You see it advancing with a constant speed, so if it is closer to the equator, its needs longer to pass meridians, due to greater spacing.

2)
A grid with equally spaced grid lines, but varying density between them. Here the density is greater near a mass, and objects advance slower trough that area. So again they need more coordinate time to reach the next grid line of their proper time in the denser area.

For the laymen: Imagine you track the same plane on a flat map of the world (http://en.wikipedia.org/wiki/Mercator_projection" ). The spacing of meridians is the same everywhere, but you see the plane advancing slower, near the equator, as if the area there was denser.

You can compare both ways here (left : way 2, right : way 1):
http://www.adamtoons.de/physics/gravitation.swf

 

[feynmann]

Also, for a hand-waving explanation, try reading

https://www.physicsforums.com/showpost.php?p=1320213&postcount=4.

First, consider a baseball that is thrown upwards. As it rises, the baseball gains potential energy and loses potential energy.

Suppose you have two identical clocks. Dig a hole straight to the centre of the Earth, leave one clock at the centre of the Earth, and hoist yourself back to the surface of the Earh, where you left the other clock.

With one eye, look in a telescope that is trained on the clock at the centre of the Earth. With the other eye, watch the clock on the surface beside you. You will see the second hand on the clock at the centre move more slowly than the second hand on the clock beside you.

Why? Because, like the baseball, photons, as they rise from the centre to the surface, lose kinetic energy. Unlike the baseball, photons always move at the speed of light, so they can't lose kinetic energy by slowing down. Photon energy is proportional to frequency, so the frequency of the light decreases as it rises.

However, frequency is like ticks of a clock. So the image of central clock that you see is slower, because the photons lost energy as the rose from the centre to your eye.

So we can explain gravitational time dilation using Newtonian gravity. We know gravitational time dilation and curvature of time are the same thing. Schutz explain that "All of Newtonian gravitation is simply the curvature of time." Isn't this reasoning circular?
So I think gravitational time dilation is more fundamental and it's better to explain it using Doppler effect in SR and equivalent principle. (Schutz book: Gravity from the Ground Up)
http://www.gravityfromthegroundup.org/pdf/timecurves.pdf

 

[A.T.]

So we can explain gravitational time dilation using Newtonian gravity.

Physics is about describing how nature works using math. It doesn't really explain anything.

We know gravitational time dilation and curvature of time are the same thing. Schutz explain that "All of Newtonian gravitation is simply the curvature of time." Isn't this reasoning circular?

Gravitational time dilation and Newtonian gravity are observed natural phenomena, which can be described using a mathematical model of curved spacetime. The observed phenomenon and the mathematical model are not the same thing.

 

[feynmann]

Physics is about describing how nature works using math. It doesn't really explain anything.

What do you think a "theory" is? e.g. <Einstein's General Theory of Relativity>
Here is the Definition of theory from the Merriam-Webster Online Dictionary
Theory: a plausible or scientifically acceptable general principle or body of principles offered to EXPLAIN phenomena <the wave theory of light>

Gravitational time dilation and Newtonian gravity are observed natural phenomena, which can be described using a mathematical model of curved spacetime. The observed phenomenon and the mathematical model are not the same thing.

You mixed up two different things: gravity and Newtonian gravity.
Gravity is observed natural phenomena. Newtonian gravity is Newton's theory of gravity.
His theory can "explain" a lot of thing that agree with experiments!

 

[A.T.]

Here is the Definition of theory from the Merriam-Webster Online Dictionary
Theory: a plausible or scientifically acceptable general principle or body of principles offered to EXPLAIN phenomena <the wave theory of light>

If you accept what physical theories do as "explaining", that's fine with with me. For me an equitation doesn't really explain anything, it just describes.

You mixed up two different things: gravity and Newtonian gravity.
Gravity is observed natural phenomena. Newtonian gravity is Newton's theory of gravity.

Yes. And if you apply the same distinction, you don't get any "circular reasoning" you mentioned before:

We know gravitational time dilation and curvature of time are the same thing

Not "the same thing":
Gravitational time dilation - observed phenomenon
Curvature of time - mathematical model

So we can explain gravitational time dilation using Newtonian gravity

No. Newtonian gravity describes only mass attraction. Curved time describes both: mass attraction and gravitational time dilation.

 

[michelcolman]

First, consider a baseball that is thrown upwards. As it rises, the baseball gains potential energy and loses potential energy.

Suppose you have two identical clocks. Dig a hole straight to the centre of the Earth, leave one clock at the centre of the Earth, and hoist yourself back to the surface of the Earh, where you left the other clock.

With one eye, look in a telescope that is trained on the clock at the centre of the Earth. With the other eye, watch the clock on the surface beside you. You will see the second hand on the clock at the centre move more slowly than the second hand on the clock beside you.

Why? Because, like the baseball, photons, as they rise from the centre to the surface, lose kinetic energy. Unlike the baseball, photons always move at the speed of light, so they can't lose kinetic energy by slowing down. Photon energy is proportional to frequency, so the frequency of the light decreases as it rises.

However, frequency is like ticks of a clock. So the image of central clock that you see is slower, because the photons lost energy as the rose from the centre to your eye.

This is a beautiful explanation, but... Wikipedia says gravitational time dilation is at its maximum at the surface of the sphere, and minimum at the center (and at infinity as well). Which makes sense, as the clock at the center does not experience any net gravity and is essentially in "free fall" while the clock at the surface is feeling an accelleration due to gravity.

So actually, the clock at the surface should be ticking more slowly, not the clock at the center. Right?

Disclaimer: I'm making my first baby steps in general relativity so I could be completely wrong.

 

[Jonathan Scott]

This is a beautiful explanation, but... Wikipedia says gravitational time dilation is at its maximum at the surface of the sphere, and minimum at the center (and at infinity as well). Which makes sense, as the clock at the center does not experience any net gravity and is essentially in "free fall" while the clock at the surface is feeling an accelleration due to gravity.

So actually, the clock at the surface should be ticking more slowly, not the clock at the center. Right?

Disclaimer: I'm making my first baby steps in general relativity so I could be completely wrong.

The Wikipedia entry on time dilation is WRONG. I've already added a note to the talk page about that just to record it. The clocks ticks slowest at the center.

 

[michelcolman]

The Wikipedia entry on time dilation is WRONG. I've already added a note to the talk page about that just to record it. The clocks ticks slowest at the center.

But I thought clocks in inertial reference frames (like the free falling apple) ticked at the same speed? The clock at the center of the Earth is definitely in an inertial reference frame.

And the total field of gravity is zero at the center of the earth. The clock doesn't "feel" any gravity, so the comparison with someone in an accellerating rocket isn't possible either.

Strange, just now when I was starting to think I understood a small part of General Relativity, it turns out I still didn't get it.

 

[A.T.]

But I thought clocks in inertial reference frames (like the free falling apple) ticked at the same speed?

The same compared to what? Never heard of this

And the total field of gravity is zero at the center of the earth.

The Newtonian field is zero, not the spacetime curvature. But that's not important. What matters is the field between the clocks.

 

[michelcolman]

The same compared to what? Never heard of this

OK, I must have misunderstood then.

I thought the equivalence principle said that the "feeling of accelleration" due to gravity was wat was causing the time dilation, just like the time dilation in an accellerating rocket. And that free-falling objects would have no time dilation due to gravity since they were not opposing it. But I guess that was wrong, then.

So, am I finally getting this correctly if I state that free-falling clocks in a field of gravity (actually, clocks that are just starting to free-fall from a stationary position, and don't have any speed yet) are ticking at the same speed as stationary clocks that are held in position inside the gravity field?

 

[A.T.]

I thought the equivalence principle said that the "feeling of accelleration" due to gravity was wat was causing the time dilation, just like the time dilation in an accellerating rocket.

Graviational time dilation is always relative: A different rate between two clocks at different positions in space. If you have to go against gravity or inertial forces due to acceleration to get from one clock to the other, then you have different clock rates.

And that free-falling objects would have no time dilation due to gravity since they were not opposing it. But I guess that was wrong, then.

Not sure if this is wrong. I might be true for two clocks free falling so close to each other, that the g-field can be considerd uniform between them. But if one clock is free falling near the Earth (or inside of it) and the second is free falling far away from the earth, they will run at different rates.

So, am I finally getting this correctly if I state that free-falling clocks in a field of gravity (actually, clocks that are just starting to free-fall from a stationary position, and don't have any speed yet) are ticking at the same speed as stationary clocks that are held in position inside the gravity field?

If the clocks are at the same position in the field of gravity, then they are identical in this case.

 

[yuiop]

The Wikipedia entry on time dilation is WRONG. I've already added a note to the talk page about that just to record it. The clocks ticks slowest at the center.

Jonathan is right about the Wikipedia entry being wrong.

But I thought clocks in inertial reference frames (like the free falling apple) ticked at the same speed? The clock at the center of the Earth is definitely in an inertial reference frame.

And the total field of gravity is zero at the center of the earth. The clock doesn't "feel" any gravity, so the comparison with someone in an accellerating rocket isn't possible either.

Strange, just now when I was starting to think I understood a small part of General Relativity, it turns out I still didn't get it.

It is a little confusing. newcomers to General Relativity are taught the equivalence principle in terms of acceleration but it turns out that gravitational time dilation is more closely related to gravitational potential than to to gravitational acceleration. As you descend towards the centre of the Earth the gravitational potential continues to fall and lower potential means slower clocks. Objects fall from high gravitational potential to lower gravitational potential.

OK, I must have misunderstood then.

I thought the equivalence principle said that the "feeling of accelleration" due to gravity was wat was causing the time dilation, just like the time dilation in an accellerating rocket. And that free-falling objects would have no time dilation due to gravity since they were not opposing it. But I guess that was wrong, then.

So, am I finally getting this correctly if I state that free-falling clocks in a field of gravity (actually, clocks that are just starting to free-fall from a stationary position, and don't have any speed yet) are ticking at the same speed as stationary clocks that are held in position inside the gravity field?

The two clocks that you describe would be ticking at the same rate until the free falling clock gains some velocity and distance relative to the stationary clock. This is a good example of one clock feeling acceleration (the stationary clock) while the other is not, yet they both experience the same amount of time dilation.

Here is the equation that makes the relationship clear:

$$t ' = {t_o \over \sqrt{1-2gm/r_o}} * {(1-2gm/r ') \over \sqrt{1-2gm/R '}}$$

where $t_o$ is the time rate according to the clock in the hand of a stationary observer located at $r_o$ and t' is the time rate of a test clock located at r' that has been released from location R'.

When the observer is located at infinity the equation reduces to:

$$t ' = t_o * {(1-2gm/r ') \over \sqrt{1-2gm/R '}}$$

When the test clock has just been released, R' is equal to r' and the equation reduces to:

$$t ' = t_o * \sqrt{1-2gm/r '}}$$

which is the same as the equation for a stationary clock located a r'.

If the test clock is dropped from the infinity while the observer remains at infinity, the equation becomes:

$$t ' = t_o * (1-2gm/r ')$$

where part of the time dilation of the test clock is due to gravitational potential at its instaneous location and part is due to its falling velocity.

An even more general equation for relative time that takes the motion of both the test clock and observer into account as well as their respective locations in the gravitational field, (presented without proof, but based on a strong hunch) is:

$$t ' = t_o *{\sqrt{1-2gm/R_o} \over (1-2gm/r_o)} * {(1-2gm/r ') \over \sqrt{1-2gm/R '}}$$

where $R_o$ is the height that the observer is released from.

All these equations are based on the radial only Schwarzschild metric which is the static vacuum solution for a non rotating gravitational body.

It is not too difficult to convert these equations into a form where the falling velocity of the test clock is explicit if anyone would prefer them presented that way.

In the case of the Earth the interior Schwarzschild solution has to used for the clock at the centre of the Earth and the equation is:

$$t ' = t_o * \left( 3 /2 * \sqrt{1-2gm/r_E} -1/2 \right)$$

which is less than the clock rate of a clock at the surface of the Earth:

$$t ' = t_o * \sqrt{1-2gm/r_E}$$

according to an observer at infinity, where $r_E$ is the radius of the Earth and a non rotating Earth is assumed.

 

[ZachN]

Time dilation occurs because we are observing things in one reference frame from another reference frame. We can ramble on about whether or not this is is accelerating or that is accelerating but it just comes down to basic idea of reference frames - time isn't really "changing". We are just traveling through a multitude of reference frames where the metric for time and space are different in each frame. Time dilation comes in when we string all these frames together as larger whole.