# Faster than speed of light due to time dilation?

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Drizy
This probably has been asked before but i had a thought about the speed of light and time dilation. First off all i know the speed of light is constant and that it is the max speed anything can be but hear me out. So let's say a jet are traveling from point a to b in space with let's say 0,8c observed from earth. The jet is traveling very fast close to the speed of light so he experiences time dilation (time goes slower for him then for earth). But from Earth he is still going 0.8c . So 1h passes for the jet and due to time dilation let's say 10h for Earth doesn't that mean he has traveled 0,8c*10h*3600s. Therefore going faster then the speed of light since the jet only took 1h for the same distance?

Mentor
Therefore going faster then the speed of light since the jet only took 1h for the same distance?
What you are describing is called rapidity <Edit: celerity, not rapidity>. It is the distance in one frame divided by the time in another frame. That quantity, although it has the same units as speed, it is not speed and is not limited to c. You can have an arbitrarily large rapidity and still be traveling slower than light.

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David Lewis, Delta2, Grasshopper and 4 others
Mentor
1h passes for the jet and due to time dilation let's say 10h for earth

This requires a velocity much closer to ##c## than ##0.8## c. The time dilation factor for ##0.8## c is only ##1.667##, not ##10##. To get a time dilation factor of ##10##, you need ##v = 0.995## c.

Delta2
Mentor
What you are describing is called rapidity. It is the distance in one frame divided by the time in another frame.

This isn't quite the usual meaning of rapidity. The usual meaning is that the rapidity ##\omega## is given by inverting the formula ##v = \tanh \omega##. For ##v = 0.995## (the correct ##v## for a time dilation factor of ##10##, per my previous post), that gives ##\omega = 2.99##, but the ratio of distance in the Earth frame divided by time in the jet frame for the OP scenario is ##9.95##. In terms of ##\omega##, that ratio is ##\sinh \omega##. In terms of the relativistic ##\gamma## and ##v##, it is ##\gamma v##.

Gold Member
What you are describing is called rapidity. It is the distance in one frame divided by the time in another frame.
It's called celerity, not rapidity!

Dale and etotheipi
Mentor
This isn't quite the usual meaning of rapidity.
Oops, yes you are right. The term is celerity, not rapidity. I will fix the post.

Drizy
Thanks for the answers! So if i understand correctly you can travel great distances relative to Earth in a short time (seen from the yet) when you are close to the speed of light. Much more then distance/time= 300 000 km/s (as seen from earth)

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2022 Award
Thanks for the answers! So if i understand correctly you can travel great distances relative to Earth in a short time (seen from the yet) when you are close to the speed of light. Much more then distance/time= 300 000 km/s (as seen from earth)
As seen from the rocket you mean? No. As seen from the rocket the rocket is not moving and the Earth is moving. The distance between Earth and the rocket's destination will be length contracted due to that speed, so the rocket will not see anything unusual.

If youuse the Earth's distance measure and the rocket's time measure then you do get the result you are stating, but it's not "seen from the rocket". It's a mix of "seen from the rocket" and "seen from the Earth".

Ren18
This probably has been asked before but i had a thought about the speed of light and time dilation. First off all i know the speed of light is constant and that it is the max speed anything can be but hear me out. So let's say a jet are traveling from point a to b in space with let's say 0,8c observed from earth. The jet is traveling very fast close to the speed of light so he experiences time dilation (time goes slower for him then for earth). But from Earth he is still going 0.8c . So 1h passes for the jet and due to time dilation let's say 10h for Earth doesn't that mean he has traveled 0,8c*10h*3600s. Therefore going faster then the speed of light since the jet only took 1h for the same distance?

Once you have fixed the speed of the jet relative to Earth, 0.8 c, you have fixed the Lorentz factor

$$γ=\dfrac{1}{\sqrt{1-0.8^2}}\approx 1.67,$$

so time dilation says that when the jet reaches its destination, in 1 h according to its clock,

$$1×1.67 = 1.67\;h$$

will have passed on Earth, and it will have travelled

$$1.67×3.6×10^3×0.8×299\,792.458\approx 1.4\;\text{billion km}.$$

There is only one thing that moves here, the jet frame relative to the Earth frame, and it moves at 80 % of the speed of light, so the Lorentz transformation says that the time interval between the two events happening at the jet location (jet at point a, jet at point b) and measured in the jet's frame (1 h) is shorter than the time interval between the same events measured in the Earth's frame (1.67 h), and the ratio between the two is 1/γ, as shown above.

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2022 Award
There is only one thing that moves here, the jet frame relative to the Earth frame, and it moves at 80 % of the speed of light, so the Lorentz transformation says that the time interval between the two events happening at the jet location (jet at point a, jet at point b) and measured in the jet's frame (1 h) is shorter than the time interval between the same events measured in the Earth's frame (1.67 h), and the ratio between the two is 1/γ, as shown above.
I think you need to be a bit careful here, because you are glossing over things that the OP seems to find confusing. The calculation you did is to determine the time measured by Earth clocks during the trip, which is ##\gamma## times the elapsed time for the jet, and hence the distance traveled in the Earth frame by the jet. That comes out to be 1.33 light hours traveled in 1.67 hours, as you say.

However, the OP's problem appears to be taking that 1.33 light hours and dividing it by the jet's clock time of 1 hour, getting a "speed" of 1.33c. As noted above, this is the celerity and not the speed. So I think it's important to carry out the equivalent calculation in an inertial frame where the jet is at rest. In this frame the distance between the start and finish points is length contracted, and is only 0.8 light hours. The jet itself is at rest, but the Earth is moving at 0.8c so the destination reaches the jet in one hour in this frame, as required. Nothing is moving faster than light in either frame.

RobertR
I am not a scientist. Can anyone explain in plain language what any of this means? If I am going near the speed of light, and time has slowed down to a minute taking a year (from an outside perspective), but a minute is still a minute from my own perspective, then am I not going faster than light from that outside perspective? Is the whole trick adjusting a minute to be relative to my own thought processes, and not to an external clock? Because I experience time as a measure of my own thought speed, and not clocks. Clocks, when they are calibrated correctly, just aid in that measurement.

Delta2
2022 Award
Can anyone explain in plain language what any of this means?
Your own body clock, and any clocks traveling with you, will always tick at one second per second as you measure them. Clocks (including body clocks) moving with respect to you will always tick slowly compared to your clocks. If you accelerate or decelerate it gets a bit complicated to describe what other clocks do - your own clocks that continue to move with you continue to tick at one second per second for you.
If I am going near the speed of light
One of the biggest mistakes you see in relativity is people saying things like this without explaining speed relative to what. There is no such thing as absolute speed - so you need to say "near the speed of light as measured by...". In this case, I think you mean as measured by an observer on Earth.
then am I not going faster than light from that outside perspective?
No. I look at how far you've gone (nearly one light year) and how long it took you (a year) and conclude that you're going at nearly lightspeed relative to me. Why would I use your clock's time? That's your measurement, not mine.

BWV
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Delta2, Grasshopper and PeroK
Arkalius
Accelerating, then decelerating constantly at 1G, you could get to the Andromeda Galaxy, a distance of 2.5 million light years, in under 30 years. On Earth 2.5 million years would have passed.

https://www.omnicalculator.com/physics/space-travel
Yeah this is an interesting result that many people find baffling. You can actually reach any location within the observable universe within a human lifetime using 1g constant acceleration halfway there and deceleration on the other half. Of course, a lot of time will have passed at your origin point, and everyone you ever knew would be long dead, but you'd make it.

Of course, there's the practical problems of how to maintain such an acceleration for such a long period of time, and how to avoid all the pitfalls of traveling through space at such high velocities. These both seem to be rather intractable complications.

David Lewis, phinds, Delta2 and 1 other person
Gold Member
Accelerating, then decelerating constantly at 1G, you could get to the Andromeda Galaxy, a distance of 2.5 million light years, in under 30 years. On Earth 2.5 million years would have passed.

https://www.omnicalculator.com/physics/space-travel
IMHO, the single coolest use for this is to travel there and back just to see what the world will look like in 2.5 million years. Most likely humanity gone, but it would still be cool to see, assuming you can deal with your entire species, and certainly everyone you know, being long gone.

Delta2
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IMHO, the single coolest use for this is to travel there and back just to see what the world will look like in 2.5 million years. Most likely humanity gone, but it would still be cool to see, assuming you can deal with your entire species, and certainly everyone you know, being long gone.
You really think that would be "cool"?

Gold Member
You really think that would be "cool"?
There are clearly immense drawbacks to it (preferably you'd do this with your loved ones with you so you wouldn't be saying goodbye forever). But at the same time, you'd be getting a sort of god's eye view of the history of your species, your planet, and your solar system.

It's a hard thing to consider, but in reality we all leave this world at some point. Everyone dies, but so few people get to experience something like this. It's like when someone goes to space. They know they could die, and never see their loved ones again, but some will still risk it. And it will happen when we start trying to colonize other planets.

This is really delving out of science, but the point I'm making is that, in the end, everyone is going to be gone long term. What you'd get out of this, beyond simply seeing the world as it will eventually be, is perhaps certainty on when goodbye is forever. Some people feel that's easier to deal with than suddenly losing someone. You'd get to truly say goodbye, with a smile and tears in your eyes, and they'd get to live their lives proudly bragging about you.

Is that worth it? I don't know. It depends on the person. You'd be giving up a great deal, but at the same time being given knowledge that no one else will ever have.

Staff Emeritus
preferably you'd do this with your loved ones
And their loved ones?
And their loved ones?

Where does it end? Or is it loved ones "all the way down"?

Gold Member
And their loved ones?
And their loved ones?

Where does it end? Or is it loved ones "all the way down"?
Lol good point. Could do like the story of Noah and extend it to three generations of one line. :D

But there would be people who would do it. And they aren’t monsters or sociopaths. They just have a different perspective.

Homework Helper
Gold Member
2022 Award
Lol good point. Could do like the story of Noah and extend it to three generations of one line. :D

But there would be people who would do it. And they aren’t monsters or sociopaths. They just have a different perspective.
Who's paying their fare? That's what I'd like to know.

Grasshopper
Homework Helper
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They are paying on time installments.

Grasshopper
Mentor
Ok, let’s please stick to the physics rather than the sociology and economics

Homework Helper
2022 Award
Isn't the physics done and dusted?

Gold Member
Accelerating, then decelerating constantly at 1G, you could get to the Andromeda Galaxy, a distance of 2.5 million light years, in under 30 years. On Earth 2.5 million years would have passed.
Note that you would lose a race with a light beam regardless of which way you do your calculations.

Ibix
Dali
Accelerating, then decelerating constantly at 1G, you could get to the Andromeda Galaxy, a distance of 2.5 million light years, in under 30 years. On Earth 2.5 million years would have passed.

https://www.omnicalculator.com/physics/space-travel
I think this is a good example of the gist the OP really was asking for. In this case it seems obvious that the crew/persons on the rocket (or whatever vessel this might be!) manages to travel much faster than light as seen from within their own rocket. Depending on whether the crew where aware of the effects of relativity or not, this can be interpreted in two ways:
1. If the crew is not aware of relativity they would conclude their own speed to be much higher than the measured speed of light (as measured on earth). They would however not overtake any light beam, and therefor also have to conclude that light speed seems to be different for each observer. They would also be very surprised to find out that time on Earth has passed extremely different from their own. (NOTE: this interpretation can never match all observations, and would eventually lead/force the crew to discover true relativity!)
2. If the crew are aware of relativistic effects, they could interpret the situation as either realising their own time is slowed down due to time dilation from their high speed relative to the earth-Andromeda rest frame (even though they can not notice it by watching their own clocks!), AND/OR that the distance to the Andromeda galaxy is shortened due to length contraction.
In relativity, both time dilation and length contraction are needed to get the numbers right. But conceptually one can always interpret these type of "faster than light"-examples as either clocks slowing down on fast moving rockets, or distances that shrink (or both!).

(Edit: added note on alt (1) clarifying that this alternative will never fit all observations, and is only the interpretation an uneducated crew would try to make!)

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PeroK and Motore
Mentor
They would however not overtake any light beam, and therefor also have to conclude that light speed seems to be different for each observer.
This shows that they could not consistently make the claim you are suggesting here. If they actually measure the speed of light they will get c. Your (1) is not a valid interpretation.

Dali
This shows that they could not consistently make the claim you are suggesting here. If they actually measure the speed of light they will get c. Your (1) is not a valid interpretation.
Yes. I should have been clearer that alt (1) is the interpretation people get into when NOT taking relativity into account. It will always clash with some measurements in reality. That is the (wrong) interpretation that people use when concluding "faster-than-light" in these cases...

(Edit: added a note in my post above clarifying this)

Dale