# Confused about traveling at the speed of light

## Main Question or Discussion Point

Let's say a space ship zooms around the Earth repeatedly at 99.99% of the speed of light for a period of time that is 50 years for observers on Earth. How much time will the passengers on board feel has elapsed after these 50 years?

And, if someone on Earth was somehow able to remotely observe what is happening on the space ship, would he see that everything is taking place in extremely slow motion?

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phinds
Gold Member
2019 Award
Let's say a space ship zooms around the Earth repeatedly at 99.99% of the speed of light for a period of time that is 50 years for observers on Earth. How much time will the passengers on board feel has elapsed after these 50 years?
Google "Lorentz transform" [and be aware that things do get a bit more complicated because you have put the spaceship into acceleration relative to Earth]

And, if someone on Earth was somehow able to remotely observe what is happening on the space ship, would he see that everything is taking place in extremely slow motion?
Yes, just as the person on the spaceship would see things on Earth taking place in extremely slow motion.

"Yes, just as the person on the spaceship would see things on Earth taking place in extremely slow motion."

Do you mean really FAST motion here?

phinds
Gold Member
2019 Award
"Yes, just as the person on the spaceship would see things on Earth taking place in extremely slow motion."

Do you mean really FAST motion here?
No, I mean what I said. Time dilation is symmetrical. BOTH parties see the other as moving very slowly.

There are literally thousands of threads on this here on PF. I suggest a forum search for "time dilation" and also Google "Twin Paradox"

NOTE: also, you should study the non-accelerated of time dilation version before you worry about a version were one party is under constant acceleration. The Twin Paradox is a good place to start.

Thanks

Ken G
Gold Member
Actually, it isn't true that both would see the other as having time move slowly. That would be true if they were zooming past each other and the distances were increasing, but it sounded like in the OP the space traveller was going "around", i.e., in a circle. That changes a lot-- the person on Earth still sees the space traveller's time as going just as slowly as before, but the space traveller cannot see Earth time as going slowly. That can only happen if there is a "relativity of simultaneity" disconnect that is constantly increasing as they separate more and more. But here, they are not separating at all, in fact at any time they can check each other's ages with no ambiguity because they are both in essentially the same place. Hence, the space traveller must agree that the Earthbound person's time is going more rapidly than their own-- there is no disagreement on this. What spoils the special relativistic "time dilation" effect for the space traveller is that they are not in an inertial frame. They must have a huge acceleration toward Earth, so would be thrown to the outside of their spaceship, as though there was a very strong gravity pointing away from Earth. That very strong gravity would be the reason they attribute time as flowing rapidly on Earth, despite Earth's relative speed to them.

It's actually a nice way to derive the general relativistic effects on time here, because you know from the need for the two people to agree on their ages that the Lorentz factor of the orbiter must determine the entire temporal differences that both people must agree on. Since the space traveller also perceives that Lorentz factor for Earth, yet must get the same answer for the increased aging of the Earthbound person, we can conclude that the general relativistic effect due to the accelerating frame is exactly double the time dilation due to the Lorentz factor. Placing the Earthbound person at the center of the Earth, and calculating the difference in effective gravitational potential due to the fictitious forces, gives you the gravitational effect on the passage of time. (To find the effective potential, imagine a spinning rigid body, and use the centrifugal fictitious force like a gravity.)

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Yes, attempting to orbit something at near light speed could get you in trouble with the law.

BvU
Homework Helper
2019 Award
Orbiting the solar system (never mind the earth) at a sensible radius would also be impossible without crushing every bone of the space travellers: $v^2/r=10$ g at $10^{12}$ km ! That's 6000 astronomical units, less than an hour per round trip (pluto has 40 AU and 248 year) -- all calculated nonrelativistically, of course .

phinds
Gold Member
2019 Award
Thanks
As you can see from the posts subsequent to mine, my focus on the fact that you were mistakenly taking time dilation as being anti-symmetrical made me seriously underestimate the issue of your exact scenario. As I advised you, best to stick with linear motion and the Twin Paradox until you have a handle on time dilation.

BvU
Ken G
Gold Member
Orbiting the solar system (never mind the earth) at a sensible radius would also be impossible without crushing every bone of the space travellers: $v^2/r=10$ g at $10^{12}$ km ! That's 6000 astronomical units, less than an hour per round trip (pluto has 40 AU and 248 year) -- all calculated nonrelativistically, of course .
Yes, it's certainly not a realistic situation! Travel near c never is, unless you take a very long time to accelerate to it.

If you traveled away from Earth at 99.9% of the speed of light in a big loop and arrived back at Earth 5 years later, the traveler would have aged 5 years and everything back on Earth would have aged about 110 years. At that speed, the traveler will have aged at 4.5% of normal time on Earth.

Ken G
Gold Member
And an important point to make is that only the Earthbound observer would regard that 4.5% factor as being consistently applied throughout the entire journey. The spaceborne traveller would at times regard the Earthbound observer as aging more quickly, and at other times more slowly, in a complicated way that would only end up agreeing with the 5 years vs. 110 years at the end of the trip. That's why the noninertial frame is complicated to use, and it is best whenever possible to stick to the inertial frames for the calculations.

Actually, it isn't true that both would see the other as having time move slowly. That would be true if they were zooming past each other and the distances were increasing, but it sounded like in the OP the space traveller was going "around", i.e., in a circle. That changes a lot-- the person on Earth still sees the space traveller's time as going just as slowly as before, but the space traveller cannot see Earth time as going slowly. That can only happen if there is a "relativity of simultaneity" disconnect that is constantly increasing as they separate more and more. But here, they are not separating at all, in fact at any time they can check each other's ages with no ambiguity because they are both in essentially the same place. Hence, the space traveller must agree that the Earthbound person's time is going more rapidly than their own-- there is no disagreement on this. What spoils the special relativistic "time dilation" effect for the space traveller is that they are not in an inertial frame. They must have a huge acceleration toward Earth, so would be thrown to the outside of their spaceship, as though there was a very strong gravity pointing away from Earth. That very strong gravity would be the reason they attribute time as flowing rapidly on Earth, despite Earth's relative speed to them.

It's actually a nice way to derive the general relativistic effects on time here, because you know from the need for the two people to agree on their ages that the Lorentz factor of the orbiter must determine the entire temporal differences that both people must agree on. Since the space traveller also perceives that Lorentz factor for Earth, yet must get the same answer for the increased aging of the Earthbound person, we can conclude that the general relativistic effect due to the accelerating frame is exactly double the time dilation due to the Lorentz factor. Placing the Earthbound person at the center of the Earth, and calculating the difference in effective gravitational potential due to the fictitious forces, gives you the gravitational effect on the passage of time. (To find the effective potential, imagine a spinning rigid body, and use the centrifugal fictitious force like a gravity.)
Can I say that in this orbiting scenario, there are 2 factors that dilate the passage of time for the observers; 1) the acceleration for the orbiter and 2) the gravitational potential for the Earthbound person?

Ken G
Gold Member
The true gravity won't play any important role, you could just neglect that. But the observer moving in a circle would reckon a dilation effect due to the apparent motion of the Earthbound observer, and a doubly great anti-dilation of that same clock due to the acceleration of the orbiting observer (I say orbiting because they are going in a circle, but at that speed, we couldn't really call it an orbit). All that is reckoned by the orbiter, of course the inertial observer sees only the time dilation of the orbiter.

The true gravity won't play any important role, you could just neglect that. But the observer moving in a circle would reckon a dilation effect due to the apparent motion of the Earthbound observer, and a doubly great anti-dilation of that same clock due to the acceleration of the orbiting observer (I say orbiting because they are going in a circle, but at that speed, we couldn't really call it an orbit). All that is reckoned by the orbiter, of course the inertial observer sees only the time dilation of the orbiter.
Ah I see. I probably misunderstood your explanation but how could the orbiting observer see a time dilation for the Earthbound observer motion and vice versa while you just said that they're not separating and thus relativity of simultaneity isn't at play here?

PAllen
2019 Award
The true gravity won't play any important role, you could just neglect that. But the observer moving in a circle would reckon a dilation effect due to the apparent motion of the Earthbound observer, and a doubly great anti-dilation of that same clock due to the acceleration of the orbiting observer (I say orbiting because they are going in a circle, but at that speed, we couldn't really call it an orbit). All that is reckoned by the orbiter, of course the inertial observer sees only the time dilation of the orbiter.
This is not true, and a simple symmetry argument establishes it. Consider the central observer emitting spherical pulses once per second. The forced circular observer will receive them at even intervals, much faster than once per second of their proper time. Analyzed in the central inertial frame, there is nothing that can possibly cause asymmetry in reception by circular observer. Since what is computed is proper time between receptions, this feature is invariant. Simple generalization establishes that the circular traveler will observe the central observer evenly elapsing time at a faster rate. Setting up a coherent coordinate system for the circular traveler is complicated - and wholly unnecessary.

[edit: I think I confused which scenario Ken claimed the rocket would see uneven time dilation for the earth. That was travel on the big loop. So ignore the claim of error, and just treat the above as a simple derivation of what the circular traveler sees.]

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Ken G
Gold Member
What's interesting about this scenario is that we always go to great lengths to point out how special relativity does not require that both observers agree on which clock is ticking more slowly, in favor of having a symmetry there. Yet here we have a case where the agreement between both observers is what must be sacrosanct, and so it is the symmetry that must go out the window. A way to preserve both the symmetry, and the agreement, would be to have two observers both moving in a circle as though they were at the ends of a bar spinning around its center. There we clearly have a symmetry between exchange of perspective, but we also must have agreement because there cannot be an accumulating relativity-of-simultaneity disconnect. So both observers must see the other's clock as ticking at the same rate as theirs. Accomplishing the motion using gravity instead of a bar, and doing so with a special type of gravity that would achieve obits that look like solid-body rotation, then tells us how to do gravitational anti-dilation of the time, because all the orbiting observers along that effectively rigid bar would agree that all the clocks tick at the same rate. This might only work in the limit of weak gravity in which the gravitational potential is mostly only affecting the time, I doubt this scenario suffices to recreate the entire metric though maybe there's some way to get a symmetry in length contraction in there somewhere too, like a constraint on the length of the circumference.

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PeroK
Homework Helper
Gold Member
Ah I see. I probably misunderstood your explanation but how could the orbiting observer see a time dilation for the Earthbound observer motion and vice versa while you just said that they're not separating and thus relativity of simultaneity isn't at play here?
Yes, relativity of simultaneity is at play here. The centripetal acceleration does not cause time dilation.

To illustrate this, I've just posted an analysis here of treating the circular orbit as a large number of small constant-velocity segments.

BvU
Yes, relativity of simultaneity is at play here. The centripetal acceleration does not cause time dilation.

To illustrate this, I've just posted an analysis here of treating the circular orbit as a large number of small constant-velocity segments.

So there IS relativity of simultaneity at play here that causes the time dilation? I thought that's only the case if the distance between 2 observers changes.

PeroK
Homework Helper
Gold Member
So there IS relativity of simultaneity at play here that causes the time dilation? I thought that's only the case if the distance between 2 observers changes.
Relativity of simultaneity and time dilation are equally fundamental aspects of relativity. Neither one causes the other. As the orbiting clock is constantly changing its IRF, it's more difficult to analyse the circular motion. One approach is to approximate circular motion as I have done. In that approach you need to use both RoS and time dilation in the orbiting clock's frame to calculate the rate of a non-orbiting clock in that frame.

There may be another method by equating the centripetal acceleration to a gravitational potential in some way. In that case, you would ignore the velocity-related time dilation and replace this with an appropriate gravitational-potential based time dilation.

Perhaps someone who knows about this can expand on it further. I'm not sure how you assign a gravitational potential to a given centripetal acceleration.

But, it's one or the other. You don't get velocity-based and acceleration-gravity-equivalent-based time dilation at the same time. However you calculate it, the answer should be asymmetric time dilation with the orbiting clock runnuing slow at a rate compatible with its orbital speed.

PS You do of course get a gravitational component of time dilation for the Earth clock. This will offset the time dilation due to the velocity of the orbiter.

The total time dilation is, therefore, a combination of a slower clock in orbit (due to its orbital speed) and a faster clock in orbit (due to its extra gravitational potential).

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So if I can understand this. Would 4 travellers 2 of them going on the same trajectory and at a speed near the C also parallel but say 1000 kms apart one going at right angles to the 2 and 1 going at 180 ° all on the same plane. How would time appear to them relative to each other?

phinds
Gold Member
2019 Award
So if I can understand this. Would 4 travellers 2 of them going on the same trajectory and at a speed near the C also parallel but say 1000 kms apart one going at right angles to the 2 and 1 going at 180 ° all on the same plane. How would time appear to them relative to each other?
That's all simple algebra and Lorentz Transforms. Check it out.

sophiecentaur
Gold Member
Let's say a space ship zooms around the Earth repeatedly at 99.99% of the speed of light for a period of time that is 50 years for observers on Earth. How much time will the passengers on board feel has elapsed after these 50 years?

And, if someone on Earth was somehow able to remotely observe what is happening on the space ship, would he see that everything is taking place in extremely slow motion?
These thought experiments are ok as far as they go but the legions of practical details can interfere with appreciating the results. Your general conclusion is in the right direction, though. If you want seriously obtainable proof of the effects on the time experienced by objects travelling near c, then look no further than what happens to muons that are produced when cosmic rays hit particles in the outer atmosphere of the Earth. These muons will decay pretty quickly (mean time 2.2μs). At the speed they are travelling (>0.99c), most of them would be expected to decay way up in the upper atmosphere. In fact, a significant number can be detected at low altitudes. This implies that their 'clocks' which tell them when to decay are running at about 1/8 of the speed of our earthbound clocks. So time (on board and measured remotely by us) has dilated by around 8 times. This wiki link gives a good description of the experimental evidence. As far as the muon is concerned, it decays 2.2μs after it woke up.
I love the fact that, according to the Wiki page:
"Since then (=the original experiments), many measurements of the mean lifetime of muons in the atmosphere and time dilation have been conducted in undergraduate experiments."

I have been trying to think of how the reciprocal effect could be 'observed' by the muons. I guess they would see the ticking of the scientists' measurement clocks going very slowly, giving the muon the impression that it is due to blow up much sooner than it expected to, according to those clock ticks.
None of this stuff is intuitive.