# Gravitational time dilation

## Main Question or Discussion Point

Timedilation seems clear when you study special relativity and read about the Hafele-Keating experiment.
Gravitational redshift seems logical when you assume that light should lose energy when it is leaving a gravitational field.
But the two seem to be contradictory to eachother.

Question 1 :Do higher frequencies of clocks/light belong to high gravitational potential or to lower ?

Question 2 :Why actually does light lose energy whereas a mass does not when it is leaving a gravitational field ?

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Orodruin
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Question 2 :Why actually does light lose energy whereas a mass does not when it is leaving a gravitational field ?
First: It is not about losing energy. In order to define energy you need to define a frame of reference and there is no unique way of comparing two events in space-time which is independent of the frame of reference. Gravitational redshift is a result for two different observers which are stationary, this defines how they will arrange and measure frequencies by defining their time direction.

Stationary observers measure the same for massive objects. A massive object will lose kinetic energy (as measured by the stationary observers) when climbing out of a potential well.

Gravitational redshift is a result for two different observers which are stationary
I believe that by this you mean stationary with respect to the mass that is responsible for spacetime curvature, i.e. hovering at constant altitude yes?

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First: It is not about losing energy.
Does that mean that the first question is waiting for somebody else ?

Maybe I should explain more : When we use a light-clock it will be easier to compare the two phenomena Timedilation and Grav.Redshift. The first one says lower frequency the second higher frequency.
Can somebody explain that to me ?

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PAllen
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You need to explain why you see a contradiction. Compared to an observer higher in a gravitational potential, lower stationary phenomena will be time dilated. That means that an oscillating charge producing what is green light per the lower observer, will be oscillating slower per the higher observer. Thus, the higher observer sees the light emitted as e.g. red. This explains the redshift rather than contradicting it.

Thus, the higher observer sees the light emitted as e.g. red.
Ok, that is clear.
Does this mean that the oscillating charge is affected by gravity, but the light it emits is not ?
So when the light is emitted it will have a constant frequency, but the observers have different frequencies.
That would mean that the light from the Sun is emitted at a lower frequency than the frequency at which we observe it and we see it redshifted.

Orodruin
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So when the light is emitted it will have a constant frequency,
You need to get the idea that light "has" a certain frequency out of your head. The frequency of the light depends on the observer and the observer needs to be at the same event as the light to measure it. Even different observers at the same event will measure different frequencies depending on their relative motion.

You need to get the idea that light "has" a certain frequency out of your head
I think it should be possible to talk about properties of objects.
When I can say that the planet Earth has a certain mass and the Sun a certain temperature, without talking about reference frames or observers, why is that different in the example above about light ? (I DID hear about special and general relativity)

Ok, I will try to formulate it more specific : "When light is emitted at the surface of the Earth would it have a constant frequency, measured by an observer on that same surface, close to the emitter ?" The observer in an orbit around the Earth would then measure a lower frequency because of his higher gravitational potential.

For example : When you prepare an experiment, you need to predict what you expect to measure.
This could mean saying that I expect the frequency of light to be constant when it crosses a gravitational field. So this is not a pure schoolbook example.

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Orodruin
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I think it should be possible to talk about properties of objects.
And it is, it is just a fact that light is not inherently associated with a frequency but the frequency depends on the observer. The equivalent of the "mass" of the Earth or the Sun is the mass of the photon, which is zero. Frequency is instead more related to the kinetic energy of a moving object, which is frame dependent.

The observer in an orbit around the Earth would then measure a lower frequency because of his higher gravitational potential.
This is not necessarily true but depends on the orbit. In addition to the gravitational time dilation, you need to take into account the time dilation from the movement. It is true that a stationary observer would measure a lower frequency, but a stationary observer has proper acceleration.

it is just a fact that light is not inherently associated with a frequency
Tnx Orodruin, I suppose you are right. I just hoped that somebody could answer my 2 questions.

PeterDonis
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I just hoped that somebody could answer my 2 questions.

Questions cannot "Just" be "answered". It depends on the person you ask them (o:

PeterDonis
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Questions cannot "Just" be "answered". It depends on the person you ask them (o:
That's not a very helpful comment. Has the discussion in this thread given you the information you wanted?

pervect
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Maybe I should explain more : When we use a light-clock it will be easier to compare the two phenomena Timedilation and Grav.Redshift. The first one says lower frequency the second higher frequency.
Can somebody explain that to me ?
Gravitational time dilation and gravitational red shift both make the same predictions. I don't see why you think they make different predictions.

The prediction that gravitational time dilation makes is that the total energy (which is composed of kinetic energy and rest energy) of a falling rock is affected in the same way that the total energy of falling light. In this context, when we say "total energy", we are not including potential energy. To put it as simply as possible, we are saying that rocks gain energy when they fall, and light does, too.

You seem to think otherwise, but it's not clear why you think otherwise. It's difficult to answer questions that make faulty assumptions . Rather than "answer the question", one has to expand the discussion to examine the assumptions underlying the question, and also to make sure that everyone is using the same fundamental defintions.

We can suspect that you may have some misconceptions about energy. Let's start with the rock. Do you agree that rocks gain energy when they fall? Do you agree that kinetic energy is energy, that a moving object has more energy than one that is not moving? Do you agree that the energy of motion of a rock (or anything, really) depends on the frame of reference you use to measure it because motion depends on the frame of reference - and therefore is not just a property of the rock, but depends on the rock and the frame of reference / observer?>

I don't see why you think they make different predictions
My problem is the following : The way people now compare the two phenomena I get the idea that a signal is created with a certain frequency and higher in the gravitational field measured with a lower frequency. That frequency shift is then caused by the different gravitational potentials of the two "observers". Orodruin explained that very clearly.
That idea looks very similar to the Doppler shift. There the frequency shift is caused by the different velocities and in our example the shift is caused by the different frequencies of the sender and receiver. So here we describe Doppler shift and Gravitational Time dilation.

But I thought Gravitational Redshift is about light flying through space and being affected by spacetime itself.
I had the idea that the frequency gradually changes between the two points, like it does by cosmological redshift.
So even when light is not created or measured in a gravitational field, it is changed by it when it crosses such a spacetime-distortion.
I can also take the example of Gravitational lensing. The light is affected (bent) by the field itself. That is not a matter of influencing the light by observers.
Isn't Gravitational Redshift working like that ?

We can suspect that you may have some misconceptions about energy
I think that the problem is not about energy. I agree with all your assumptions. Although it would be a good idea to start another topic on the subject "why did we introduce the term potential (gravitational) energy and what exactly do we mean by that ?"

That's not a very helpful comment. Has the discussion in this thread given you the information you wanted?
No, and sorry about the last comment that was indeed not very helpful. I tried to explain what bothers me in a reply on Pervect's comment.

we are saying that rocks gain energy when they fall, and light does, too
Do you mean that light accelerates like the rock or the light is getting blueshifted or maybe both ?

Timedilation seems clear when you study special relativity and read about the Hafele-Keating experiment.
Gravitational redshift seems logical when you assume that light should lose energy when it is leaving a gravitational field.
But the two seem to be contradictory to eachother.

Question 1 :Do higher frequencies of clocks/light belong to high gravitational potential or to lower ?

Question 2 :Why actually does light lose energy whereas a mass does not when it is leaving a gravitational field ?
Well seen! Okun actually wrote a paper about that issue, as it has to do with a misconception.
- http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.286.5360
- http://arxiv.org/pdf/physics/9907017v2.pdf

1. A higher clock frequency corresponds to a higher gravitational potential. In view of that, Einstein predicted that light emitted from the Sun will be redshifted, because that light will be emitted at a lower frequency than light from a corresponding source on Earth.

2. The frequency difference at observation is fully accounted for by the difference in emission frequencies; there is no place for an additional redshift due to "loosing energy". Note that it cannot be otherwise, as the number of cycles must be conserved at constant distance in a dispersion free vacuum.

Thank you harryLin, I think this answers my questions, but I will probably need a year to fully understand those articles (o:

Thank you harryLin, I think this answers my questions, but I will probably need a year to fully understand those articles (o:
You're welcome

In fact it's just one article, but the explanation is IMHO needlessly complex. The essence of it is not really difficult when keeping track of wave cycles and by sticking to a single reference, for example a reference system far away from gravitational fields.
I elaborated in a number of posts in an earlier thread with some simple numerical examples, as the discussion was hindered by ambiguity of words and definitions: #3 [URL='https://www.physicsforums.com/threads/time-dilation-in-the-field-interpretation-of-gr.800631/#post-5030307']#5 etc. up to #52.

PeterDonis
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That idea looks very similar to the Doppler shift.
This apparent analogy is highly misleading, and will only lead to confusion. There is a way to generalize how we model "observed frequency shift" so that it covers both of these possibilities, but it requires looking at everything geometrically and hence dropping both "natural" interpretations (difference in "potential" for gravitational frequency shift, and difference in velocity for Doppler frequency shift).

pervect
Staff Emeritus
Do you mean that light accelerates like the rock or the light is getting blueshifted or maybe both ?
The light doesn't change it's speed, but it gains energy. Digressing a bit to the quantum picture, the number of photons in the light beam doesn't change, but the energy per each photon does, as $E = h \nu$, where $\nu$ is the frequency. Thus when you blueshift the light, you increase the energy per photon without changing the number of photons, winding up with a greater amount of energy.

Unlike the rock, the light does not change it's speed, the local speed of light is always "c". But like the rock, it does gain energy when it falls.

The light doesn't change it's speed, but it gains energy. Digressing a bit to the quantum picture, the number of photons in the light beam doesn't change, but the energy per each photon does, as $E = h \nu$, where $\nu$ is the frequency. Thus when you blueshift the light, you increase the energy per photon without changing the number of photons, winding up with a greater amount of energy.

Unlike the rock, the light does not change it's speed, the local speed of light is always "c". But like the rock, it does gain energy when it falls.
More precisely, the locally measured energy (with a locally calibrated detector) of the light is increasingly higher when the downward propagating light is detected at increasingly lower altitude.

PAllen
2019 Award
In relativity, it is common that a given set observations can be given different explanations (often, but not always, related to what coordinates you use for modeling). You can't say one is right or wrong, but there may be different pedagogic value to different descriptions (about which people may still disagree). A trivial example is "why do muons created in the upper atmosphere reach the ground in large numbers?" Is it due to time dilation or distance contraction? Depends on coordinates used (earth based or muon based).

In the case of the phenomenon of 'gravitational redshift': light emitted as one frequency at the bottom of a tall building is received at the top as a lower frequency

There are at least 3 explanatory frameworks that can be made consistent. You get into real trouble only if you try to mix them. I should also say, that I agree with most of the posts above as to which is pedagogically most useful (as I described in my post #5). However, the OP confusion in part came from mixing two alternate explanatory frameworks, and thinking they should be additive, so I hope to clarify and separate these frameworks.

To help motivate each, I will formalize a thought experiment as follows:

- From the ground of a tall building, we emit a pulse of light at a known frequency and also a nearly (rest)mass-less body at vanishingly close to c (imagine the classical analog of a ball of neutrinos; call this an n-blob). The n-blob is emitted with a known (substantial) KE (kinetic energy).

1) Gravitational Time Dilation Explanatory Framework

A clock at the top of the building runs faster than a clock at the bottom, according to a stationary observer at either location. The same is true for any physical process related to time. As a result, the light pulse emitted at one frequency at the bottom (per a bottom observer), is considered to be emitted at a lower frequency by a top observer, and is received at this lower frequency at the top. The same goes for the energy of the pulse. Note that all of the energy of light is kinetic energy, because it has no rest mass. The speed of the light pulse remains c whether measured at the top or the bottom.
Looking at the n-blob, you need to understand the intimate connection between time and energy, alluded to by Pervect earlier. [You can crudely motivate this by imagining a ball bouncing between two close walls. A higher observer must see this 'clock' running slow compared to an adjacent observer, thus it the ball must be moving slower, and have less KE. Then, in order to have consistent local behavior at different altitudes, all energy (and mass) must scale the same way as time rate.]. Thus, per the higher observer, the n-blob is emitted with less energy than as measured by the lower observer, and this is the energy later detected by the higher observer. The speed of the n-blob will be only infinitesimally different at top compared to the bottom, and infinitesimally different from c. For such a blob, even a 100 fold change in energy would be associated with an undetectable difference in speed.
Note that potential energy has not been mentioned in this framework.
Note that for a slow moving body, at an altitude where the time related scaling of total energy brings the total energy down to the rest energy, the body cannot go any higher (without extra impulse of some kind). Since light has no rest energy (it is all kinetic energy), there is no lower bound on energy scaling, thus light always escapes. The n-blob behaves essentially indistinguishably from light.

2) Potential Energy Explanatory Framework

The notion of gravitational potential energy is introduced. It is consistent with (1) in that the differences in measurement by stationary observers are referenced to a standard at infinity. KE (as measured by stationary observers) is considered exchangeable with potential energy. For gravity and relativity, we must add the notion that total energy plays the role that m plays in a pure Newtonian potential (that is, that potential difference acts on total energy(/c^2) rather than just (rest) mass, though the energy gained or lost is KE). Seeing that this is a different packaging of the same effect as the time dilation explanation clarifies that you would never try to combine these effects - they are describing the same thing in different ways.
That the n-blob or a normal body may be considered to exchange KE for potential energy as it climbs a gravity well is certainly non-controversial. The n-blob introduces the fact that you must consider potential difference as acting on total energy /c^2, not rest mass, or you would find (incorrectly) that the n-blob loses essentially no KE climbing a gravitational well. For a normal body, KE/c^2 is so small compared to m, that this difference is not noticed.
Some say it is 'wrong' to apply this explanation to a light pulse. Yet, the math of the n-blob in GR is, in the limit, indistinguishable from a light pusle. It must, therefore, be no more wrong to say the light pulse exchanges KE for PE than it is for the n-blob. In the case of light, there are many pedagogic advantages to focusing on time dilation and the classical wave picture of light, but comparing a light pulse to an n-blob establishes that it can't be 'wrong' to use a potential energy framework. You don't even need to bring photons into the picture.
A normal body may reach a height where its KE is zero, and then it must fall. Light just gets asymptotically closer to an 'energy at infinity', as does an n-blob (for all practical purposes).

3) And now for something completely different: coordinates based on a free fall observer.

Caveat: this really works only over a small region (like our tall building), where tidal gravity is not significant. For the sake of argument, we consider the tall building suspended on struts to more easily imagine a free fall frame whose origin is coincident with the building bottom at emission time, and is momentarily at rest at that time.
A free fall frame agrees with the time dilation framework that neither light, nor an n-blob, nor a normal body, change KE after emission. However, it finds that the unchanged energy is the one measured by the building bottom observer, and offers a completely different explanation for why the building top observer measures lower energy in all cases. It is simply that the building top observer is moving in this frame by the time it detects each emission.
For light, the motion of the building top at time of reception compared to the bottom at time of emission, leads to Doppler red shift.
For material bodies, it is simply that their KE is less because they are measured by an observer moving in the same direction as the bodies. For the n-blob, you must use relativistic formulas, and find that the result is (again) indistinguishable from the light pulse Doppler computation for change of frequency (energy).
For a 'normal' body, it is seen that building's speed may soon catch up to the body's, at which point the body starts getting closer to the floor.
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Don't fret about which is correct, rejoice that that the same set of observations can be understood from multiple points of view.

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Thus, if developed correctly, the potential energy explanatory framework is perfectly consistent with the gravitational time dilation explanatory framework; it is then the same "point of view".
It should be noted though that this is only true if the space-time in question is stationary, i.e. if it admits a time-like Killing vector field, otherwise the concept of "potential energy" in itself becomes meaningless, or at least very difficult to define.