Gravity, time and interplanetary communication

[Brunolem33]

Imagine that we had discovered an habitable planet, Planet X, located 10 light years away from Earth, and that we had managed to establish a small colony there.

Now imagine that because of a much stronger gravity than on Earth, time on Planet X would run 5 times slower than on Earth: 1 year on Planet X = 5 years on Earth

Someone sending a radio signal from Earth to Planet X would have to wait 20 years (10 + 10) to receive an answer.

But what about the other way round?

How much time would elapse for someone located on Planet X?

More generally, does gravity and thus the flow of time affect the way time/distances are perceived in light years?

 

[PeroK]

Imagine that we had discovered an habitable planet, Planet X, located 10 light years away from Earth, and that we had managed to establish a small colony there.

Now imagine that because of a much stronger gravity than on Earth, time on Planet X would run 5 times slower than on Earth: 1 year on Planet X = 5 years on Earth

Someone sending a radio signal from Earth to Planet X would have to wait 20 years (10 + 10) to receive an answer.

But what about the other way round?

How much time would elapse for someone located on Planet X?

More generally, does gravity and thus the flow of time affect the way time/distances are perceived in light years?

First, you've been watching too much science fiction. The variations in proper time on the surface of different planets is negligible. One of the reasons that relativity took so long to discover was that its effects are largely negligible in terrrestrial situations.

For example, if we took a "planet" that was the size of the Earth, but the mass of the Sun, then the difference in proper time would about 0.01%.

 

[Brunolem33]

Understood, but the planet was just used as a (bad) example in order to illustrate the question regarding the flow of time in two very different situations.

What about a neutron star? would gravity affect time enough there?

And if yes, what would be the answer to my question regarding a communication between Earth and this neutron star (even though life, or even a radio, would not be welcome there...it is just theoretical)?

 

[fresh_42]

@Brunolem33:
As @PeroK already has mentioned, some basics about SR are needed to answer these kind of questions. It's therefore difficult to explain where you got wrong.

Temporary closed for moderation.

 

[Nugatory]

Reopening this thread in the relativity forum, where we may be able to better help the original poster.

 

[PeroK]

Understood, but the planet was just used as a (bad) example in order to illustrate the question regarding the flow of time in two very different situations.

What about a neutron star? would gravity affect time enough there?

And if yes, what would be the answer to my question regarding a communication between Earth and this neutron star (even though life, or even a radio, would not be welcome there...it is just theoretical)?

Frequency of messages sent is a measure of time, so there would be a discrepancy in message frequencies.

 

[PeterDonis]

What about a neutron star? would gravity affect time enough there?

No. It is impossible to have a stable object with a time dilation factor of 5 on its surface. The reason is a theorem called Buchdahl's Theorem, which says that an object can only be in a stable equilibrium if its radius is at least 9/8 of the Schwarzschild radius corresponding to its mass. The time dilation factor at the minimum radius is therefore

$$
\frac{1}{\sqrt{1 - \frac{r_s}{r}}} = \frac{1}{\sqrt{1 - \frac{8}{9}}} = \frac{1}{\sqrt{\frac{1}{9}}} = 3
$$

So no stable object can have a time dilation factor of more than 3 at its surface, as compared to infinity (and Earth's time dilation factor at its surface is close enough to that at infinity that it makes no difference here). Neutron stars are the most compact stable objects we know of, and they don't come very close to the limit; AFAIK the surface radius of the most compact neutron stars is about twice the Schwarzschild radius, which would give a time dilation factor of $\sqrt{2}$, or about 1.4.

The only way to get larger time dilation factors is to get close enough to the horizon of a black hole. For a non-rotating black hole, you have to use rocket thrust to "hover" that close to the horizon, which is not realistic. For a hole that is rotating rapidly enough, you can find free-fall orbits close enough to the horizon to give large time dilation factors. But those orbits will be unstable--small perturbations will either fling you out of orbit towards infinity or drop you into the black hole.

 

[PeterDonis]

How much time would elapse for someone located on Planet X?

One-fifth of the time elapsed on Earth clocks (assuming that Planet X and Earth are at rest relative to each other). That's what the time dilation factor means.

does gravity and thus the flow of time affect the way time/distances are perceived in light years?

It affects the elapsed time on clocks. It doesn't affect distances. Gravity is not the same thing as changing inertial frames in SR; there is no "gravitational length contraction", and the time dilation is not symmetric.

 

[Dale]

Hmm, I think that we are dismissing @Brunolem33 question too quickly. He knows that there is gravitational time dilation and also that there is a finite round trip transit time for a radar pulse. He is asking if the round trip transit time scales the same way as the time dilation.

I think that the answer is yes, but I have never actually done that calculation.

 

[PeterDonis]

He is asking if the round trip transit time scales the same way as the time dilation.

I think that the answer is yes

It is. That was the point of the first part of my response in post #8.

 

[Dale]

It is. That was the point of the first part of my response in post #8.

But does the time dilation factor mean the same thing as the round trip radar delay? My intuition says that they are equal, but it doesn't seem like they are the same thing.

 

[PeterDonis]

does the time dilation factor mean the same thing as the round trip radar delay?

The time dilation factor is the ratio of elapsed times along the two worldlines (that of the Planet X observer and that of the Earth observer) between events that are simultaneous. (We are assuming that the two observers are at rest relative to each other.) The round trip radar delay picks out particular simultaneous events (emission and reception on the sending worldline, and the corresponding simultaneous events on the other worldline, using simultaneity as defined by the family of spacelike hypersurfaces that are orthogonal to both observers' worldlines). That's what I was referring to in the first part of post #8.

My intuition says that they are equal

It's straightforward to prove this mathematically for any static spacetime. (Note that just the fact that it is static is enough; you don't have to use any particular properties of Schwarzschild spacetime.)

 

[Dale]

Yes, I see that now. The round trip radar time to any fixed distance is a light clock, so it must dilate like any clock.

 

[pervect]

Imagine that we had discovered an habitable planet, Planet X, located 10 light years away from Earth, and that we had managed to establish a small colony there.

Now imagine that because of a much stronger gravity than on Earth, time on Planet X would run 5 times slower than on Earth: 1 year on Planet X = 5 years on Earth

The math doesn't work out for that simple a scenerio. But if you have planet X orbiting a rotating black hole, it's vaguely feasible that you could have something roughly similar to your hypothetical question. I don't have the book, but see for instance Kip Thorne's https://www.amazon.com/dp/0393351378/?tag=pfamazon01-20 for the details.

Someone sending a radio signal from Earth to Planet X would have to wait 20 years (10 + 10) to receive an answer.

But what about the other way round?

How much time would elapse for someone located on Planet X?

More generally, does gravity and thus the flow of time affect the way time/distances are perceived in light years?

II've not seen this worked out in detail, but what I think should happen is that if the Earth sends out a signal once a year, planet X will recive a signal every 1/5 a year. Now planet X is orbiting the black hole, but I think we can apply conservation of energy arguments in this particular case (which is rather tricky in GR in general, but this case should be stationary and have a time-like Killing vector, so the arguments should work). This would mean a constant doppler shift between signals sent and signals received.

So if the Earth sent out a signal once every 5 years, planet X would receive a signal roughly once every year, and vica-versa. Talking about "travel times" gets complicated because it assumes a particular simultaneity convention, it's much more straightforwards to give the observed doppler shifts, and note that while the travel time won't be exactly constant, it should be reasonably close to being constant, given that the diameter of the plaent's orbit is small compared to 5 light years.

So , to answer the original question as modified (with planet X orbiting a rotating black hole ala "Interstellar", I think it would be 10 Earth years proper time for an exchange of messages, but only 2 "planet X" years for a round trip exchange. Unless I've made an error, of course - the book I mentioned might have a further discussion, but I don't own a copy, so I'm working this out on my own.

 

[Stavros Kiri]

It affects the elapsed time on clocks. It doesn't affect distances. Gravity is not the same thing as changing inertial frames in SR; there is no "gravitational length contraction", and the time dilation is not symmetric.

Together with 'proper time' isn't there a 'proper length' ?

 

[ShayanJ]

Together with 'proper time' isn't there a 'proper length' ?

Proper time is the length of the world-line of an observer. Its called time because the world-line of an observer is time-like. But there is no observer with a space-like world-line, i.e. everything moves slower than light!

 

[BenAS]

So , to answer the original question as modified (with planet X orbiting a rotating black hole ala "Interstellar", I think it would be 10 Earth years proper time for an exchange of messages, but only 2 "planet X" years for a round trip exchange. Unless I've made an error, of course - the book I mentioned might have a further discussion, but I don't own a copy, so I'm working this out on my own.

This got me curious about something, if the people on planet x measured the distance to the sun using parallax, what result would they get? Different (shorter) from a parallax measurement of planet x made from Earth I would guess?

 

[PeterDonis]

this case should be stationary and have a time-like Killing vector, so the arguments should work

It will work provided that each light signal is sent/received by Planet X in exactly the same point of its orbit, relative to Earth (the easiest point to pick is the one where Planet X is at its closest point to Earth, i.e., Planet X and Earth lie on the same radial line).

Together with 'proper time' isn't there a 'proper length' ?

Yes, but it doesn't work the same as in SR.

if the people on planet x measured the distance to the sun using parallax, what result would they get?

I'm not sure offhand. I would think there would be some effect due to the bending of light by the black hole's gravity, but I'm not sure how large it would be since the light coming from the Sun to Planet X would not be passing close to the black hole in between.

 

[Ibix]

I'm not sure offhand. I would think there would be some effect due to the bending of light by the black hole's gravity, but I'm not sure how large it would be since the light coming from the Sun to Planet X would not be passing close to the black hole in between.

Possibly relevant - the diagram at the bottom of this post. It shows a spray of incoming light rays at two different altitudes. It's for someone hovering above a non-rotating black hole, not orbiting a rotating one, but still. It's the rays that are entering the diagram vertically that are of interest, since they could come from a distant star at rightangles to a baseline across an orbit. The rays drawn on that diagram arrive at ten degree intervals. A quick look suggests that two observers hovering at 10M (five Schwarzschild radii) on opposite sides of the hole would see the distant star shift about ten degrees. Naive triangulation would suggest they're practically sitting on the star...