# Gravitational Time Dilation - Confused

Hey everyone, recently I watched a discovery program about time travel. I understood time travel from the view point of traveling at/close to the speed of light, however, I am completely confused as to how time travel is possible using gravity.

I know that it has to do with gravitational potential, but under that logic, one could say that as gravity increases, so does the speed at which objects fall, ergo, making them age slower. (Assuming that objects can fall at/close to the speed of light)

I just don't understand the whole concept of Gravitational Time Dilation, can anyone please shed some light on it?

## Answers and Replies

tiny-tim
Science Advisor
Homework Helper
Welcome to PF!

Hey valdar! Welcome to PF!
Hey everyone, recently I watched a discovery program about time travel. I understood time travel from the view point of traveling at/close to the speed of light, however, I am completely confused as to how time travel is possible using gravity.

It isn't!

Using wormholes possibly, using gravity definitely not.

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Hey everyone, recently I watched a discovery program about time travel. I understood time travel from the view point of traveling at/close to the speed of light,

First it should be made clear that the "time travel" when travelling at close to the speed of light is just the relative slowing of the moving clock relative to the clock of a stationary observer. There is no going backwards in time and murdering you grandparents or whatever else time travellers have the urge to do. There is no changeing the order of a sequence of events that have already happened (causalty). There is just differential ageing.

however, I am completely confused as to how time travel is possible using gravity.
I know that it has to do with gravitational potential, but under that logic, one could say that as gravity increases, so does the speed at which objects fall, ergo, making them age slower.

Although the speed of a falling clock does have some influence on the slow down of the proper time of the falling clock it is not the crux of the matter. A perfectly stationary clock low down in a gravitational potential well, runs slow relative to clock higher up.

(Assuming that objects can fall at/close to the speed of light)
Yes, they can. An object released from infinity aproaches the local speed of light as it aproaches the event horizon of a black hole. From the point of view of an observer higher up the velocity of the falling object (and the coordinate speed of light) aproaches zero as it aproaches the event horizon.

I just don't understand the whole concept of Gravitational Time Dilation, can anyone please shed some light on it?
There is a lot of ground to cover here and I have a feeling this is going to be a long thread. You can get started by googling the "equivalence principle".

Thank you for replying. My biggest spheel is the following:

https://www.physicsforums.com/showpost.php?p=1320213&postcount=4
as they rise from the centre to the surface, lose kinetic energy.
but as they fell from the surface to the centre, they gained kinetic energy

The vary same photons that ended up being "decelerated" while moving from the clock to the observers eye are "accelerated" when they move from a light source (lets say at the same distance as the observer themselves) to the clock.

So if the distances are the same, the deceleration is equal to the acceleration, there shouldn't be any shifts. Where am I wrong?

Thank you for replying. My biggest spheel is the following:

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

but as they fell from the surface to the centre, they gained kinetic energy

The vary same photons that ended up being "decelerated" while moving from the clock to the observers eye are "accelerated" when they move from a light source (lets say at the same distance as the observer themselves) to the clock.

So if the distances are the same, the deceleration is equal to the acceleration, there shouldn't be any shifts. Where am I wrong?

You would be right if the source of the photons in George's hole was a torch in the hand of the observer at the top of the hole. On the way down to the clock at the bottom of the hole they gain energy (higher frequency) and then after being reflected off the bottom clock they would lose the same amount of energy and arrive back at the top with the same frequency that they were emitted with. I think in George's example the clock is the source of the photons. (Imagine the hands painted with glow in the dark luminous paint)

Sorry for the double post, but I --THINK-- I get it.

Let's say you have a room. In that room you have a ball bouncing between the two walls. Every bounce is one second.

You get two of the rooms. You put one room on earths surface, and another in the centre of the earth.

For the room at the surface, the balls bounce is accelerated by gravity.
For the room at the centre, the balls bounce is not accelerated by gravity.

Observing the centre balls bounces from the surface, it will seem slower, and vice versa, is this about right?

*thumbs up* except with light its not about physical acceleration but rather a path difference caused by curved space time. Also a freqency change (gravitational redshift). But it has much the same effect, in this example.

Sorry for the double post, but I --THINK-- I get it.

Let's say you have a room. In that room you have a ball bouncing between the two walls. Every bounce is one second.

You get two of the rooms. You put one room on earths surface, and another in the centre of the earth.

For the room at the surface, the balls bounce is accelerated by gravity.
For the room at the centre, the balls bounce is not accelerated by gravity.

Observing the centre balls bounces from the surface, it will seem slower, and vice versa, is this about right?

It doesn't just seem slower, it IS slower. The slow down is not due to lack of being accelerated by gravity either. Imagine the balls are "super balls" that lose no energy with each bounce and they are bouncing from side to side horizonatally so that gravity has no direct influence on its bounce speed. A digital device counts each bounce of the balls. When one of the ball rooms is lowered to the centre of Earth and and brought slowly back to the surface again, the room that was at the centre would have counted less bounces, but both balls will be bouncing at the same speed when back together at the surface.

Ok then there is another issue.

Not being in a gravity field makes your internal clock run slower.THIS POINT IS WRONG. Being in the middle of a strong field makes your internal clock run slower
Traveling at the speed of light makes your internal clock run slower.

The faster you go the more you weigh, the more you weigh the more mass you have, making you, yourself have a gravitational field. But then how are you both able to have your internal clock run slow due to traveling fast, but at the same time, have it run fast by being the creating factor of a gravity field?

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sylas
Science Advisor
Ok then there is another issue.

Not being in a gravity field makes your internal clock run slower.
Traveling at the speed of light makes your internal clock run slower.

Um... it is the clock that is deep inside a gravitational field that runs more slowly.

There's a great story on the web of a scientist-dad who took his kids on a camping trip up the mountains, along with some atomic clocks. See Clocks, Kids, and General Relativity on Mt Rainier. Just because I think the pictures are hilarious, here's a shot of plan A: backpack your atomic clock, compared with plan B: use the car and drive up a mountain.
(<-- Plan A. Plan B -->)

They used plan B, and also left some clocks back home for the subsequent comparison.

The clocks up the mountain run faster. By going up the mountain with his kids, this Dad got to spend an extra 22 nanoseconds with his kids that he'd have missed by staying home. As he says: It was the best extra 22 nanoseconds I've ever spent with the kids.

The faster you go the more you weigh, the more you weigh the more mass you have, making you, yourself have a gravitational field. But then how are you both able to have your internal clock run slow due to traveling fast, but at the same time, have it run fast by being the creating factor of a gravity field?

Actually, the faster you go, the more energy you have ... as measured by someone who remains at rest! For you, in motion, there's no extra energy or extra gravity, involved.

Cheers -- Sylas

Um... it is the clock that is deep inside a gravitational field that runs more slowly.

Actually, the faster you go, the more energy you have ... as measured by someone who remains at rest! For you, in motion, there's no extra energy or extra gravity, involved.

Cheers -- Sylas

From an outside observer, you would be in the middle of a gravitational well, traveling at close to the speed of light then, right?

If so, then that's what I was looking for.

Wow the world is really subjective than I originally thought

JesseM
Science Advisor
From an outside observer, you would be in the middle of a gravitational well, traveling at close to the speed of light then, right?
Why do you say you'd be traveling at close to the speed of light? In the most common coordinate system to use for spherical bodies like planets (Schwarzschild coordinates), if you're at a constant distance from the center then your speed is zero.

Why do you say you'd be traveling at close to the speed of light? In the most common coordinate system to use for spherical bodies like planets (Schwarzschild coordinates), if you're at a constant distance from the center then your speed is zero.

Sorry, let me explain that better, my original question was that as a mass accelerates, its weight increases, with an increase in weight there would be an increase in gravitational pull towards the object.

Originally I had a mistake in my assumptions. Correctly, an object traveling very quickly would be both deep in a gravitational well (due to its own increase in mass, gravity) and it would have time pass by slower, and due to the fact that it is traveling close to the speed of light, it would also have time pass by slower.

sylas
Science Advisor
From an outside observer, you would be in the middle of a gravitational well, traveling at close to the speed of light then, right?

I don't think so; the association of gravity and the additional energy of a moving object is not that simple. In fact, special relativity works all by itself in your example.

If you have a clock on a massive object (large rest mass) which is moving (relative to you) at near to light speed, then there are two components in the time difference: the velocity part, that can be obtained with special relativity, and the gravitational part, for which you need general relativity.

Wow the world is really subjective than I originally thought

Indeed! It's worth mentioning, however, that rest mass is not subjective. It is an invariant.

If you have two observers moving relative to each other at high speed, then their observations are symmetrical. From the perspective of either one, the other clock is slower.

This symmetry is broken if one of the observers shifts their inertial frame, by changing velocity, and that's why there's no actual paradox with any of the travelling twins variations.

In the case of gravitational time dilation, there's no symmetry between the observers. Each observer agrees that the clock in the kitchen is deeper into a gravitational field and running more slowly than the clock up on Mt Rainer. You can have observers on the mountain, and in the kitchen at home, in direct communication with each other, and agreeing without ambiguity that the kitchen clock runs slow.

Cheers -- Sylas

If you have a clock on a massive object (large rest mass) which is moving (relative to you) at near to light speed, then there are two components in the time difference: the velocity part, that can be obtained with special relativity, and the gravitational part, for which you need general relativity.
Cheers -- Sylas

This was what I was trying to get at.

Thank you :)

From the information in this thread, I think the conceptualized spacetime material is actually drawn wrong.

An "Empty" (objectless) area of space is has no gravity exerted upon it (minute, but very close to 0).
The centre of any object in space (Excluding black holes) has no gravity exerted upon it either.

If gravity is the culprit that bends the spacetime continuum, shouldn't it be equal between the core of an object, and empty space?

see the attached picture - Please excuse my terrible MS paint skills

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sylas
Science Advisor
From the information in this thread, I think the conceptualized spacetime material is actually drawn wrong.

An "Empty" (objectless) area of space is has no gravity exerted upon it (minute, but very close to 0).
The centre of any object in space (Excluding black holes) has no gravity exerted upon it either.

If gravity is the culprit that bends the spacetime continuum, shouldn't it be equal between the core of an object, and empty space?

see the attached picture - Please excuse my terrible MS paint skills

No, the conventional picture is the one that is more correct.

You can show this by thinking of the gravitational blue shift of a light of a fixed reference frequency being shined down a deep hole to a detector at the center of the Earth.

There is a blue shift observed when you shine the light down the height of the tower at Harvard. Put another way; the atomic clock at the top of the tower is running faster than the one at the bottom.

Now imagine digging into the basement. Do you think there will be more, or less blueshift observed?

There will be more blueshift observed, of course; and that's true because the light is moving deeper into the gravitational well. So it goes, all the way down to the center.

This is getting back to being subjective again. A clock at the center of the Earth runs more slowly than one at the surface. You figure out how much more slowly NOT by measuring the gravitational acceleration in each location; but by considering a world line between the two locations... which involves moving deeper and deeper into a gravitational well.

Cheers -- sylas

From your example, there is a specific distance between the top and bottom of the tower. During that distance there is a specific amount of gravity excreted.
If you dig into the basement, more gravity is excreted (due to increase in distance), increasing the blue shift.

If you dig deep enough, gravity will become weaker and weaker, decreasing the -amount- that the blue shift happens by.

It will still happen, it will still increase, but at a less and less value, which will be 0 at the centre then?

Is that right?

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sylas
Science Advisor
From your example, there is a specific distance between the top and bottom of the tower. During that distance there is a specific amount of gravity excreted.
If you dig into the basement, more gravity is excreted (due to increase in distance), increasing the blue shift.

If you dig deep enough, gravity will become weaker and weaker, decreasing the -amount- that the blue shift happens by.

It will still happen, it will still increase, but at a less and less value, which will be 0 at the centre then?

Is that right?

Yes, I believe so. That is, the rate of change of blue shift goes to zero at the center of the Earth. But it's more and more blue shift all the way down.

Inside a large hollow cavity at the center of the Earth, observers distributed around the cavity would all see their clocks running at the same rate, and they would all see the same difference between their clock, and a clock at the surface.

Cheers -- sylas

A.T.
Science Advisor
From your example, there is a specific distance between the top and bottom of the tower. During that distance there is a specific amount of gravity excreted.
If you dig into the basement, more gravity is excreted (due to increase in distance), increasing the blue shift.

If you dig deep enough, gravity will become weaker and weaker, decreasing the -amount- that the blue shift happens by.

It will still happen, it will still increase, but at a less and less value, which will be 0 at the centre then?

Is that right?

Yes, there was a recent thread about this:
https://www.physicsforums.com/showthread.php?t=308904
Gravitational time dilation is not a local effect, but a clock rate ratio between two distant points in space. It is determined by the spacetime curvature between the two points, and can occur even if spacetime is locally flat at both points.

From your example, there is a specific distance between the top and bottom of the tower. During that distance there is a specific amount of gravity excreted.
If you dig into the basement, more gravity is excreted (due to increase in distance), increasing the blue shift.

If you dig deep enough, gravity will become weaker and weaker, decreasing the -amount- that the blue shift happens by.

It will still happen, it will still increase, but at a less and less value, which will be 0 at the centre then?

Is that right?

Nope, it's wrong. Gravitational time dilation is not as closely connected to gravitational acceleration as you think and has more to do with gravitational potential. For example a clock at the centre of the Earth will be running slower than a clock at the centre of the Moon because the Earth has more mass and is in a deeper gravitational well. You can think of the rubber sheet analogy. A large bowling ball representing the Earth puts a deeper dent (gravitational well) in the rubber sheet than the dent made by a cricket ball representing the Moon. The gravitational acceleration at the centre of the Earth and at the centre of the Moon is the same (zero) but the clocks at the centres of the two bodies are NOT running at the same rate. The blue shift is increasing the deeper you go and the larger the mass of the body and does NOT go towards zero. If the body is sufficiently massive and its mass has collapsed towards a radius of 2GM/c^2 then a clock at the centre will stop (and the blue shift become infinite.) as the body becomes a black hole. That is the limiting case.

<EDIT> I think I have misinterpreted your statement above. My statement above is more applicable to this earlier statement of yours:

An "Empty" (objectless) area of space is has no gravity exerted upon it (minute, but very close to 0).
The centre of any object in space (Excluding black holes) has no gravity exerted upon it either.

If gravity is the culprit that bends the spacetime continuum, shouldn't it be equal between the core of an object, and empty space?

The time dilation at the core of an object is not equal to the time dilation of empty space.

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Look at it this way, if you have a stationary massive object in space with a tunnel (from one end to the other) in the middle of it, say a planet. As the light is traveling down the tunnel it is blue shifted until the centre of the object. From the centre of the object to the other side of the tunnel the light is red shifted, overall, there would be no shift, would there?

See the attached image.

2nd part. I understand that in the centre of the earth a clock would run slower than on the surface, how come?

Gravitational time dilation is not a local effect, but a clock rate ratio between two distant points in space. It is determined by the spacetime curvature between the two points, and can occur even if spacetime is locally flat at both points.

So then the speed of time is measured by how high or low we are on the spacetime continuum? relatively speaking of course.

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sylas
Science Advisor
Look at it this way, if you have a stationary massive object in space with a tunnel (from one end to the other) in the middle of it, say a planet. As the light is traveling down the tunnel it is blue shifted until the centre of the object. From the centre of the object to the other side of the tunnel the light is red shifted, overall, there would be no shift, would there?

That's correct. But remember, any "shift" is always dependent on an observer; not an absolute property of the light.

You can only say the light coming out of the tunnel is "redshifted" as a comparison between what is observed in the center, and what is observed half way out.

2nd part. I understand that in the centre of the earth a clock would run slower than on the surface, how come?

Because that's how massive objects bend spacetime.

So then the speed of time is measured by how high or low we are on the spacetime continuum? relatively speaking of course.

I don't think that's a meaningful statement, as phrased. "high and low on the spacetime continuum"?

The passage of time is measured by clocks. A clock has a "world line", and "proper time" along a world line is what is measured by a clock with that world line. Relativity is the theory that allows you to infer what will be measured by a clock, given its world line.

Cheers -- sylas

That's correct. But remember, any "shift" is always dependent on an observer; not an absolute property of the light.

In the above example, as drawn out in MS paint, how would you see the light shifted inside the planet? you see the light leave a source, travel into, and through the planet, and then out the other side..... what would you see happen? wouldnt you see both shifts

wait.... in order for you to see both shifts that means that at every point in time during that line light is given off from that line which then travels towards you. wow. based on that youd see something completely different.

That didn't work. Lets say that you are measuring the blue/red shifts of light. One measuring device is located at the beginning (before the blue shift) one in the middle (when the blue shift becomes red shift) and at the end (at the end of the red shift). In that case wouldnt you be able to "sense" the shifts using the measuring tools?

I don't think that's a meaningful statement, as phrased. "high and low on the spacetime continuum"?

The passage of time is measured by clocks. A clock has a "world line", and "proper time" along a world line is what is measured by a clock with that world line. Relativity is the theory that allows you to infer what will be measured by a clock, given its world line.

Cheers -- sylas

Meaning, clocks only measure how high we are in the spacetime continuum, in other words if we are on a heavy object, or not. At a high point on the continuum clocks would run at a specific speed, at a low point they would run at another one, how far off am i?

In the above example, as drawn out in MS paint, how would you see the light shifted inside the planet? you see the light leave a source, travel into, and through the planet, and then out the other side..... what would you see happen? wouldnt you see both shifts

No single observer would be able to see the entire light path and all the shifts. You would require a series of observers arranged along the path. Imagine the test beam is wide and they have small detectors that can sample the beam without blocking it. Later they can compare measurements and compare them with what GR predicts. Using the calculations of GR they can predict what they would measure at some point they have not yet tested and then check if the prediction is correct. After predicting and checking that the equations are correct for several random points they can build up confidence in the predictions and determine what the frequency is at any point using only maths without requiring an infinite number of observers. Oh, and yes they would see both shifts.

That didn't work. Lets say that you are measuring the blue/red shifts of light. One measuring device is located at the beginning (before the blue shift) one in the middle (when the blue shift becomes red shift) and at the end (at the end of the red shift). In that case wouldnt you be able to "sense" the shifts using the measuring tools?
Yes, as above.

Meaning, clocks only measure how high we are in the spacetime continuum, in other words if we are on a heavy object, or not. At a high point on the continuum clocks would run at a specific speed, at a low point they would run at another one, how far off am i?

continuum is a vague word. If you mean that at a high gravitational potential clocks run faster relative to clocks at a low gravitational potential... then yes.

2nd part. I understand that in the centre of the earth a clock would run slower than on the surface, how come?

Science has very few answers for why the universe is the way it is or how it works the way it does (what mechanism). It can only tell you what you would expect to measure given certain parameters and offer some guiding principles learnt from observation and experience. One of those guiding principles in GR is the equivalence principle and another is the constant speed of light as measured locally.

So we can not say WHY clocks in at a low gravitational potential run slower than clocks at higher gravitational potential anymore than we can state WHY objects fall from a high potential to a low potential. (not yet anyway without a complete gravitational theory of everything). There is some connection between the two observations and it might be possible to claim at some level that objects fall from a high gravitational potential to a lower gravitational potential because time runs slower at lower gravitational potential (or vice versa).

When studying relativity you have to careful to note when you are talking about relative or coordinate or local or proper measurements. For example the statement that the speed of light is always constant is only true if you only mean the local speed of light. The statement that clocks run at different speeds is only true if you mean the relative rate rather than the proper rate. The proper rate of a clock is always one second per second but to claim that clocks do not run at different rates without making it clear that you specifically mean only proper clock rates is misleading just as it is misleading to claim the speed of light is always constant without making it clear you are only talking about the local speed of light. As you go deeper into a gravity well the relative speed of light gets slower, the relative speed of clocks gets slower and the relative length of rulers get shorter and the consequence of all that is that any observer always measures the local speed of light to be the same.

Another surprise is that the frequency of a rising photon does not actually change! The observer higher up with the faster clock measures the frequency of the light as decreasing, merely because he is making the measurement with a faster clock! When you think about it, it makes sense. The energy of a photon is determined by its frequency. If its frequency was really getting lower then you have to ask where has all the missing energy gone? Objects usually shed energy by emitting photons, but photons do not shed energy by emitting more photons. Another mechanism used in Newtonian physics is that objects give up there energy to a gravitational field but in GR the gravitational field is not considered to store energy in the classical way. Therefore a rising or falling photon neither gains nor loses energy, but locally it appears to.

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DrGreg
Science Advisor
Gold Member
Another surprise is that the frequency of a rising photon does not actually change! The observer higher up with the faster clock measures the frequency of the light as decreasing, merely because he is making the measurement with a faster clock! When you think about it, it makes sense. The energy of a photon is determined by its frequency. If its frequency was really getting lower then you have to ask where has all the missing energy gone? Objects usually shed energy by emitting photons, but photons do not shed energy by emitting more photons. Another mechanism used in Newtonian physics is that objects give up there energy to a gravitational field but in GR the gravitational field is not considered to store energy in the classical way. Therefore a rising or falling photon neither gains nor loses energy, but locally it appears to.
I'm afraid that doesn't really make sense. The only frequency a photon has is the frequency an observer measures with a clock. Different observers measure different frequencies, so to say "the frequency stays the same" is wrong if two observers measure different frequencies at different heights. Frequency is an observer-dependent concept, as are time, distance, energy, etc. If two different inertial observers measure different energies there's no need to explain where the energy has gone because they are measuring in different frames.

And if you choose to measure a rising photon at two different heights in a single accelerating frame, the photon's loss of energy (gravitational red-shift) is accounted for as a gain in potential energy. (That logic would apply even to accelerating frames in the absence of gravity.)

I'm afraid that doesn't really make sense. The only frequency a photon has is the frequency an observer measures with a clock. Different observers measure different frequencies, so to say "the frequency stays the same" is wrong if two observers measure different frequencies at different heights. Frequency is an observer-dependent concept, as are time, distance, energy, etc. If two different inertial observers measure different energies there's no need to explain where the energy has gone because they are measuring in different frames.

I am talking more about the intrinsic physical frequency or energy of the photon. I agree that observers at different heights would measure different frequencies and that is the conventional view but I am looking deeper than that, as I always like to get a feel for what is really physically happening by deduction and logic. Meaurements often involve calculations rather than being direct measurements. For example when we measure velocity we measure the distance and record the elapsed time to cover that distance and then divide the distance by time. To measure distance we require two spatially separated observers and two clocks. In a gravitational field the two clocks at either end of a vertically orientated ruler are running at different rates. If we calculate the distance by timeing how long it takes a light signal to travel from the top of the ruler (point A) to the bottom of the ruler (point B) we need a start time at A and a finish time at point B. It is not sensible to make a measurement with two clocks running at different rates, so we should choose one clock as the master clock and synchronise the other clock by speeding it up or slowing it down so that they both run at the same rate. For example observer A could set his clock to send a test signal at one second intervals down to observer B. Observer B could speed up his clock so that the signals from A arrive at one second intervals according to (speeded up) clock B. Observer B could also send signals up to A at one second intervals (on his speeded up clock) and A could confirm the signals are arriving at one second intervals. The two clocks are now synchronised and a sensible measurement of distance and velocity can now be made. With this arrangement it would be confirmed that the speed of light is slower, lower down in the gravitational field (as predicted by the Schwarzchild metric) and that the frequency of light does not really change as it moves vertically in a gravitational field. We KNOW clocks run at different rates at different heights, so it is a very blinkered approach to ignore that FACT when comparing other measurements at different heights that have a time component.

And if you choose to measure a rising photon at two different heights in a single accelerating frame, the photon's loss of energy (gravitational red-shift) is accounted for as a gain in potential energy. (That logic would apply even to accelerating frames in the absence of gravity.)
Potential energy in the gravitational context is a Newtonian concept. In GR, the concept of a rising photon giving up its energy to be stored as potential energy in a classical gravitational field is very questionable. See this discussion by Baez: http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

Here is something else to consider. I shine a household torch at a black hole for a few seconds. The photons from the torch are blue shifted as they fall towards the black hole and the frequency of the photons goes to infinity as they approach the event horizon of the black hole. Have really added infinite energy to the black hole by shining a torch at it for a few seconds? I think not. It seems more likely that if the torch emitted one Joule of energy, then the black hole eventually absorbs one Joule of energy. Therefore the energy of a photon does not change as it falls. Although GR can use many coordinate systems and in some of them it appears that energy is not conserved, it seems that we can always find a coordinate system where energy is conserved. That is something that is worth bearing in mind when you get junk mail offering to sell you a machine for \$10,000 that will produce infinite free energy "as proven by General Relativity".

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sylas
Science Advisor
I am talking more about the intrinsic physical frequency or energy of the photon. I agree that observers at different heights would measure different frequencies and that is the conventional view but I am looking deeper than that, as I always like to get a feel for what is really physically happening by deduction and logic.

A photon really does not have any intrinsic physical frequency. Its frequency is relative to an observer; always. There's nothing else.

Here's a thought experiment for you. How could you tell whether or not *we* are all inside a large cavity, within a very large massive object? Can you use any "intrinsic" properties of photons to recognize that they are all blueshifted, down here in the hole in the middle of a super-planet?

The reality is -- you can't tell. There's nothing intrinsic about the frequency of a photon that lets you tell whether you are observing it at the bottom of a gravitational well, or outside the well. The redshift or blueshift is always and only a comparison of two observers; not an intrinsic property of a photon.

Cheers -- sylas

DrGreg
Science Advisor
Gold Member
I am talking more about the intrinsic physical frequency or energy of the photon. I agree that observers at different heights would measure different frequencies and that is the conventional view but I am looking deeper than that, as I always like to get a feel for what is really physically happening by deduction and logic.
But you are just arbitrarily choosing one observer as your reference observer and then transforming other observations to the local frame of the reference observer. You're entitled to do this, but I'm entitled to choose another observer as my reference and come to a different conclusion.

If you're looking for some frame-invariant notion of frequency, the only answer is zero, which doesn't help much. (Zero because it's proportional to energy; the "rest energy" of a photon is proportional to its "rest mass" which is zero. Or just let $v \rightarrow c$ in the Doppler shift formula.)

Potential energy in the gravitational context is a Newtonian concept. In GR, the concept of a rising photon giving up its energy to be stored as potential energy in a classical gravitational field is very questionable. See this discussion by Baez: http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html
In general, energy is a problem in GR. I'm no expert in this, but I understand that in some specific circumstances it is nevertheless possible to come up with a sensible definition of energy that is conserved. The link that you provided includes these words:

"In certain special cases, energy conservation works out with fewer caveats. The two main examples are static spacetimes and asymptotically flat spacetimes."

"The Schwarzschild metric is both static and asymptotically flat, and energy conservation holds without major pitfalls."​

I haven't just made up potential energy in GR. In Schwarzschild coordinates it is, for a hovering object,

$$\frac{mc^2}{\sqrt{1 - 2GM/rc^2}}$$​

See Woodhouse, N M J (2007), General Relativity, Springer, London, ISBN 978-1-84628-486-1, pp. 100-101.

Here is something else to consider. I shine a household torch at a black hole for a few seconds. The photons from the torch are blue shifted as they fall towards the black hole and the frequency of the photons goes to infinity as they approach the event horizon of the black hole. Have really added infinite energy to the black hole by shining a torch at it for a few seconds?
No, because, in the frame of a distant hovering observer the photons would also lose an infinite amount of potential energy as they fell (taking an infinite time to do so). (Other frames would disagree, as energy is a frame-dependent concept.)

Here is something else to consider. I shine a household torch at a black hole for a few seconds. The photons from the torch are blue shifted as they fall towards the black hole and the frequency of the photons goes to infinity as they approach the event horizon of the black hole. Have really added infinite energy to the black hole by shining a torch at it for a few seconds?

If you shine the torch from deep space (no gravity around you) then yes that is what happens.

However, if you are standing on earth and you do that, the photons would first be RED shifted as they leave earth (Earths gravity is pulling it back)... then once they leave Earths gravity well there is no shift. When the light hits the black holes gravity well then there would be a blue shift.
This would be visible to ALL observers, its the interaction of light and gravity isnt it?

I haven't just made up potential energy in GR. In Schwarzschild coordinates it is, for a hovering object,

$$\frac{mc^2}{\sqrt{1 - 2GM/rc^2}}$$​

See Woodhouse, N M J (2007), General Relativity, Springer, London, ISBN 978-1-84628-486-1, pp. 100-101.

There is somethng wrong with that equation. Is there a missing minus sign or should it be

$${mc^2}*{\sqrt{1 - 2GM/rc^2}}$$ ?​

As it stands, the equation you have posted suggests the potential energy is lowest at infinity. That is the exact opposite of Newtonian gravitational potential energy. Perhaps you could quote some of the text from Woodhouse to put the equation in context?

No, because, in the frame of a distant hovering observer the photons would also lose an infinite amount of potential energy as they fell (taking an infinite time to do so). (Other frames would disagree, as energy is a frame-dependent concept.)

The small "m" in the equation you posted is rest mass. A photon does not have rest mass so either a photon always has zero potential energy or that equation does not apply to photons. Can you show an equation that applies to photons seeing as that is what were talking about? Without that, your claim that a photon loses infinite potential energy (and donates inifinite energy to a black hole?) is unsupported.

"In certain special cases, energy conservation works out with fewer caveats. The two main examples are static spacetimes and asymptotically flat spacetimes."

"The Schwarzschild metric is both static and asymptotically flat, and energy conservation holds without major pitfalls."​

Fine words, but can someone demonstrate that energy is conserved with a numerical example? For example let us take a one kilogram mass that is stationary at infinity and let it fall to say a radius of 4GM/c^2. Can someone show a complete energy balance taking the terminal velocity, kinetic energy, momentum energy and potential energy into account such that energy is conserved?

(Use units such that c=1, m=1, G=1 and assume the mass of the black hole (M) =1000m)

This equation might be useful although I am not sure if it applies in a gravitational field:

Total energy = $$\sqrt{p^2c^2+m^2c^4}$$

where $$p = \gamma mv$$

and $$\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$$

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DrGreg
Science Advisor
Gold Member
kev, you are right, there is something wrong with my last post.

First, I mistyped Woodhouse's formula, which is actually

$$\sqrt{1-2M/r}$$​

for the PE of a unit mass (in geometrised units). Clearly there is a problem applying that to a zero mass.

I will go away and have a deeper think about this. When I've collected my thoughts I will probably start a new thread about gravitational PE in GR, as we are straying from this thread's topic.

In the meantime, take a look at http://people.maths.ox.ac.uk/~nwoodh/gr/index.html [Broken], on which his later book was closely based. Lecture 12, Section 12.2 (pages 54-55), is the relevant section I was quoting.

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kev, you are right, there is something wrong with my last post.

First, I mistyped Woodhouse's formula, which is actually

$$\sqrt{1-2M/r}$$​

for the PE of a unit mass (in geometrised units). Clearly there is a problem applying that to a zero mass.

I will go away and have a deeper think about this. When I've collected my thoughts I will probably start a new thread about gravitational PE in GR, as we are straying from this thread's topic.

Cheers, I look forward to the new thread. Exploring PE in some more detail will be interesting.

If you shine the torch from deep space (no gravity around you) then yes that is what happens.

However, if you are standing on earth and you do that, the photons would first be RED shifted as they leave earth (Earths gravity is pulling it back)... then once they leave Earths gravity well there is no shift. When the light hits the black holes gravity well then there would be a blue shift.
Essentially correct (using unsynchronised clocks). The red shift leaving Earth will be microscopic compared to the blue shift that approaches infinite as the photon approaches the event horizon of the BH.

A photon really does not have any intrinsic physical frequency. Its frequency is relative to an observer; always. There's nothing else.

Here's a thought experiment for you. How could you tell whether or not *we* are all inside a large cavity, within a very large massive object? Can you use any "intrinsic" properties of photons to recognize that they are all blueshifted, down here in the hole in the middle of a super-planet?

The reality is -- you can't tell. There's nothing intrinsic about the frequency of a photon that lets you tell whether you are observing it at the bottom of a gravitational well, or outside the well. The redshift or blueshift is always and only a comparison of two observers; not an intrinsic property of a photon.

Cheers -- sylas
I agree that in the cavity you would not measure any intrinsic redshift or blueshift. I was talking about relative measurements between two different observers at different potentials and intelligently interpreting the result to draw valid conclusions. I will post more tomorrow on time time dialtion and photon frequency when I have more time.

An extended thought experiment to illustrate some interesting aspects of gravitational time dilation and photon frequency.

A ring is constructed around a black hole. It has a very large radius and is centred on the black hole. A smaller secondary ring is constructed with a radius a little larger that the event horizon of the black hole. One observer (A) is placed on the larger ring and a second observer (B) is placed on the smaller ring. Each observer has a standard clock, a small standard ruler, a long tape measure, a light signalling device and an ideal fibre optic cable. They each measure the local speed of light using their ruler and clock and note that the local speed of light is c (299,792,458 m/s). They each measure the circumference of their respective rings and time how long it takes a light signal to circumnavigate the ring inside an ideal fibre optic cable layed along the ring and note that the circumference of ring as measured by the tape measure is consistent with the circumference of the ring as calculated by timing how long it takes a light signal to go around. In other words nothing unusual is noted. A vertical ladder is constructed joining the two rings. A third observer (C)with their own standard clock and ruler checks the measuring devices against those of observer A to make sure they are correctly calibrated and then descends very slowly down the ladder and compares the transported clock and ruler against those of observer B. Again nothing unusual is noted. According to observer C the clocks of A and B are running at the same rate using C's direct local comparison method.

The initial conclusions from the measurements taken so far are:

Gravity has no effect on clocks.
Gravity has no effect on rulers.
Gravity has no effect on the speed of light.

These initial conclusions are naive and it appears that nothing interesting happens. The project funder is furious. He has spent millions on this experiment and this is the best this bunch of idiots can come up with? He issues further instructions. First they compare of the frequency of signals sent up and down and confirm that the frequency appears to change according to the gravitational gamma factor $$\gamma$$. Next they send signals up and down at one second intervals and conclude that the lower clock is running slower by a factor of y relative to the higher clock. They speed up the lower clock so that it is synchronised with the higher clock and now note that the speed of light horizontally is slower by a factor of y relative to the speed of light higher up. They measure the vertical distance of the connecting ladder using a rulers layed end to end and note that the measured vertical distance is a lot longer than would be assumed by calculating the distance using r = circumference/2/pi. Detailed examination reveals that rulers orientated vertically length contract by a factor of $$\gamma$$ also.

When they piece all the information together the final conclusions are:

Lower clocks slow by a factor of $$\gamma$$ relative to clocks higher up.
Lower vertical rulers contract by a factor of $$\gamma$$. relative to rulers higher up.
Horizontal rulers do not length contract.
The horizontal speed of light is slower by a factor of $$\gamma$$ relative to the speed higher up.
The vertical speed of light is slower by a factor of $$\gamma^2$$ relative to the speed higher up.
The relative frequency of light does not change as photons rise or fall.
The energy of a photon relative to a photon higher up is constant.
The relative wavelength of light lower down is shorter by a factor of $$\gamma$$ relative to light higher up.

They note that the equation c = wavelength*frequency still holds but in a gravitational field it is frequency that is constant and not the speed of light c when considered in a relative way.

Relative length, relative time and the relative speed of light all tend to zero as the event horizon is approached. Indeed it can be shown that photons hover at the event horizon. See: http://www.mathpages.com/rr/s7-03/7-03.htm

JesseM
Science Advisor
A vertical ladder is constructed joining the two rings. A third observer (C)with their own standard clock and ruler checks the measuring devices against those of observer A to make sure they are correctly calibrated and then descends very slowly down the ladder and compares the transported clock and ruler against those of observer B. Again nothing unusual is noted. According to observer C the clocks of A and B are running at the same rate using C's direct local comparison method.
If this third observer were to synchronize his clock with A, descend slowly down the ladder, stay with B for a while and then slowly acend back up the ladder, he would note that his clock had fallen behind A's, so he'd know that some form of time dilation was taking place even if locally his clock always matched the rate of whoever he was next to at that moment.
kev said:
Relative length, relative time and the relative speed of light all tend to zero as the event horizon is approached.
Only if by "relative" you mean "relative to the Schwarzschild coordinate system", which in GR is just one of an infinite number of equally valid coordinate systems for dealing with a black hole spacetime. One could also presumably come up with a coordinate system where "relative length, relative time and the relative speed of light all tend to zero" as you approach some radius from the center of the Earth--in both cases these statements have no deep physical significance, they are just an artifact of a particular choice of coordinate system.
kev said:
Indeed it can be shown that photons hover at the event horizon. See: http://www.mathpages.com/rr/s7-03/7-03.htm
What that page is saying is that if a photon is directed outward in a radial direction by an event exactly on the event horizon, then the photon will hover there (in Kruskal-Szekeres coordinates its diagonal worldline would coincide with the diagonal representing the event horizon in these coordinates). It is certainly not saying that photons in general hover at the horizon, a photon moving inward will cross it just fine.