# Slowing of time under gravity

I am not calling this gravitational time dilation because that is a relativistic effect due to the equivalence principle. Now imagine two light clocks (a clock that ticks due to light) are placed above the surface of the earth. One is placed at sea level and one very high. You might be beginning to think that I am copying Einstein but please bear with me. Consider the following image: The left part of the image shows the light clock placed closer to the earth's surface and the right part shows the clock placed very high. We do not need exact height as we are not doing any calculations. One light ray is emitted from the left end of the clock on the left of the image. Now since it is closer to the earth, it will bend more than the light ray in the other clock and take a longer time to reach the receiving end. The above clock (which is the clock on the right part of the image) is away from the earth and experiences less gravity. Therefore the light ray will bend less and reach the receiving end faster. Just imagine that the clocks are fixed in position and are not moving. They are only ticking when the light ray reaches the other end. Even GR predicts that a clock will run slower in high gravity but doesn't actually give a reason.

Won't this also lead to the slowing of time in a higher gravitational area (where the force of gravity is more)? Can this be an actual reason for gravitational time dilation or not?

NOTE The clocks are not falling. The light ray inside them will still bend as it is not being held in position by anything. Only the clocks are fixed. How can a light ray be fixed? So do not think that the clocks are falling.

• To find the right answer it have to be taken in attention how the clocks could be synchronized. The start signals to the clocks and the coming back signals are also under the influence of gravity. Where has to be placed the position so that the start signals arrive to booth clocks at the same moment? Only in this case the right difference of the back coming signals can be measured. – HolgerFiedler Jun 23 '14 at 18:45
• Well that would lead to complication for this question. I want to find the answer for my question first and then we could build upon the question. Lets just assume the start signal is emitted at the same time for both the clocks. And then we could get into the pratical aspect. – rahulgarg12342 Jun 23 '14 at 18:59
• Now I could imagine how to carry out this experiment. Yoa are in between the two clocks and send to them the start signal. After getting back the stopp signals you have to move down or up instead the signals came back at the same time. Now you double the way for the lights rays in both clocks. If you get now the stopp signals back with some difference you won. – HolgerFiedler Jun 24 '14 at 5:05
• Well its good you thought how to do it. But the question here already assumes that you know how to do it. We just want a theoretical description of the result we should get and then we could practically test those predictions. – rahulgarg12342 Jun 24 '14 at 5:07
• You are not asking about the clocks frequency. You are asking about the time for the rays between to mirrors. This is clever. But to find an answer you have to note that distance between the mirrors maybe changes. And once again have attention about the synchronization. Without this you cant find out does one device runs faster the overs. – HolgerFiedler Jun 24 '14 at 19:19

## 3 Answers

I am not calling this gravitational time dilation because that is a relativistic effect due to the equivalence principle. ... But GR predicts it only through accelerating frames and thus bringing in relativity. I do not see any relativity in my case. Its just spacetime curvature.

What you have is gravitational time dilation. But even if you don't call it such, in any static, curved spacetime with time dilation between stationary clocks, at least some of the clocks are necessarily accelerated. A static spacetime has a privileged timelike direction in which geometry remains the same, and its metric tensor $g$ can therefore be put in the following form: $$\mathrm{d}s^2 = -e^{2\phi}\,\mathrm{d}t^2 + h_{ij}\,\mathrm{d}x^i\,\mathrm{d}x^j\text{,}$$ where $\phi$ is a function that may depend on any spatial coordinates $x^i$ ($i = 1,2,3$), but not on $t$. A clock stationary in this spacetime will have four-acceleration components $a^\alpha = g^{\alpha\beta}\phi_{,\beta}$. If there is some time dilation between stationary clocks at different positions, then $\phi$ is not constant with respect to the spatial coordinates, and so at least some of the partial derivatives $\phi_{,\beta}$ don't vanish: therefore, some (perhaps all) of the stationary clocks are necessarily accelerated.

Furthermore, any theory in which test particles in freefall take geodesics in a curved spacetime automatically satisfies the weak equivalence principle.

GR explains it in the following way:- Suppose you are falling from space into the earth, in a higher position your speed is not very high and thus you experience less time dilation (as time dilation directly depends on your velocity) while when you get closer to the surface, your speed becomes higher and you experience more time dilation.

No. Consider two observers stationary in a gravitational field, one on top of another, separated by a height along which gravity doesn't change much (so we can approximate the gravitational field as uniform). Since they are accelerated, the situation is equivalent to having them outside a gravitational field but in a vertically accelerating rocket.

Let's say the rocket is transparent. Then a horizontal light beam entering the accelerating rocket will be see to curve downward in the rocket's frame. Thus it must do so in the case of a gravitational field as well. With a few light beam reflections, this situation is identical your setup.

Won't this also lead to the slowing of time in a higher gravitational area (where the force of gravity is more)?

Yes.

Can this be an actual reason for gravitational time dilation or not?

Sort of. Instead of passing the light beam horizontally, it would have been much simpler to pass it vertically. Suppose two observers stationary in a gravitational field, one on top of each other, separated by a height small enough so that gravity doesn't change much over it, pass each other light pulses at regular intervals (according to their reckoning). This situation is then equivalent to doing the same outside a gravitational field but in a vertically accelerating rocket. Then it becomes obvious that the bottom is receiving signals at a rate different from which the top is sending them, as the bottom is catching up to the light pulses while they are in transit.

Again: this isn't really specific to GTR. The argument would be virtually identical in any curved, static spacetime, because stationary observers in such a spacetime are accelerated. We can then consider the situation in a local inertial frame and everything else follows as above. The key relativistic insight is simply that gravitational fields disappear in inertial frames.

Even GR predicts that a clock will run slower in high gravity but doesn't actually give a reason.

The usual presentation of GTR's reasoning regarding gravitational time dilation involves a setup morally equivalent to yours, except most often presented in a simpler way, because having light rays go in the same direction as the acceleration is easier. It has nothing to do with speed gained while falling as you seem to think.

You have written that the bottom is receiving signals at a different rate. But I would like to remind you that gravity does not cause any speed of change in light when light is vertical. Please explain.

The only difference is the orientation of the light clock. The speed of light is the same in every inertial frame, and in the inertial frame instantaneously comoving with the light clock (free-falling from rest), both ends of the light clock accelerate upward. Thus, the bottom accelerates toward a downward pulse and the top accelerates away from an upward pulse, so the distance those pulses travel is different.

As for how the situation looks in the stationary frame, the coordinate speed of light is different at the top of the light clock is different from the coordinate speed of light at the bottom.With an explicit radial coordinate, the the above metric (without $c=1$ units) is $$\mathrm{d}s^2 = -e^{2\phi}\,c^2\,\mathrm{d}t^2 + h_{rr}\,\mathrm{d}r^2 + \{\text{other spatial terms}\}\text{,}$$ so that vertical light has $$\left|\frac{\mathrm{d}r}{\mathrm{d}t}\right| = c\frac{e^{\phi}}{\sqrt{h_{rr}}}\text{.}$$ Since top part has proper time $\tau_\text{top}$ with $\mathrm{d}\tau_\text{top} = e^{\phi_\text{top}}\,\mathrm{d}t$ and the bottom part has proper time $\tau_\text{bot}$ with $\mathrm{d}\tau_\text{bot} = e^{\phi_\text{bot}}\,\mathrm{d}t$, usually with $\phi_\text{top}\neq\phi_\text{bot}$, they will disagree with the amount of time vertical light pulses take to traverse the same $r$-coordinate interval.

By the way, GTR predicts that for weak fields (appropriate near our Earth), $h_{rr} \approx e^{-\phi}$ and $\phi\approx\Phi/c^2$, where $\Phi$ is the Newtonian gravitational potential. But the above keeps the discussion general to any static spacetime, whether it agrees with what GTR predicts for the situation or not.

I did not get how is the two observers case similar to one in a rocket outside space as you have said.

In a local inertial frame, the two observers accelerate upward. The condition that the gravitational field does not change much over the relevant height means that we can consider their acceleration to be the same. That's the acceleration of the rocket, but you can forget about the rocket if you wish and just consider the same acceleration in a local inertial frame.

I am not calling this gravitational time dilation because that is a relativistic effect due to the equivalence principle.

Notice one thing: Equivalence principle says that gravity down is equivalent to acceleration up. Now, from this Einstein moved on to formulate the General Relativity Theory, where gravitational acceleration is replaced with space(time) curvature. So in classical mechanics you have gravity as acceleration and in GR you have gravity as curvature. So if we assume this definition as true, we have gravitational time dilatation in both cases

Anyway, back to your question:

The above clock (which is the clock on the right part of the image) is away from the earth and experiences less gravity. Therefore the light ray will bend less and reach the receiving end faster.

Won't this also lead to the slowing of time in a higher gravitational area?

Looks like it :) That's how GR explains time dilatation "graphically".

If 1 second is defined as a period it takes for the light ray to go from emitter to receiver, and the light ray of the higher-placed clock takes shorter for to travel from one to the other, then - as you say - this clock would tick faster than the lower-placed clock. Therefore 1 second on Earth is longer time as compared to 1 second above Earth. So the higher-placed clock goes faster and the lower-placed clock goes slower.

• Thats exactly what I said. Even GR predicts that the higher placed clock is faster. I am just asking could my question be the reason for why it actually happens? And by a higher gravitational area I meant the place where the force of gravity is more. I think you interpreted it as height. – rahulgarg12342 Jun 23 '14 at 20:01
• OK, I misread what you wrote and thought of higher position. I will correct it right away. – bright magus Jun 23 '14 at 20:05
• But GR predicts it only through accelerating frames and thus bringing in relativity. I do not see any relativity in my case. Its just spacetime curvature. GR explains it in the following way:- Suppose you are falling from space into the earth, in a higher position your speed is not very high and thus you experience less time dilation (as time dilation directly depends on your velocity) while when you get closer to the surface, your speed becomes higher and you experience more time dilation. – rahulgarg12342 Jun 23 '14 at 20:35
• But your clocks aren't falling, are they? They maintain their distance relative to the surface of Earth and to reach other. – bright magus Jun 23 '14 at 20:45
• NO the clocks are not falling. The light ray is inside the clock. But even if the clocks are not falling the light ray inside them will bend. – rahulgarg12342 Jun 24 '14 at 4:33

Won't this also lead to the slowing of time in a higher gravitational area (where the force of gravity is more)? Can this be an actual reason for gravitational time dilation or not?

Yes, that's exactly what happens. For this reason GPS need to take into account the different times in the orbit and on the earths surface.

Add: Concerning the comments about possible experiments, that's also a way to measure the effect.