# If a muon travelling fast can “extend” its lifespan due to relativistic effects, would the muon see itself travelling faster than light?

In other words, if I put a bomb set for 2.2 microseconds and send it out at .99c, would it travel further than (.99c x 2.2 microseconds)? And if it does, like muons do, wouldn’t the bomb be able to tell that it has gone faster than light for the 2 microseconds it was alive?

• "If a laboratory traveling fast can "extend" its life span due to relativistic effects, would the laboratory see itself traveling faster than light?" Remember, in the reference frame of the muon the muon is not traveling at all. The two reference frames are equivalent; anything you want to ask the muon you can as well ask the laboratory, where you are warm and dry and have a nice desk. Aug 23, 2021 at 21:47

Yes, it would travel further than $$0.99c \times 2.2 \,\mu\text{s}$$. It would travel $$\beta \gamma c \times \Delta t$$ because of time dilation. This does not mean however, that the muon or the bomb see the surroundings moving faster than the speed of light. This is because in the frame of the muon distances are relativistically contracted. So if you have some rod that you use to measure the distance the muon has passed, then the muon sees this rod contracted, so that if the muon decayed at the end of the rod, it would still find $$\beta = 0.99$$.

• Ah, I see I still have some things to learn, thank you for the answer Aug 21, 2021 at 17:13
• For those who don't know, $\gamma$ is the Lorenz factor. I am not sure what $\beta$ is - it looks like it is the percent decimal factor of the speed of light constant $c$ .
– john
Aug 23, 2021 at 5:20
• @john I think that is pretty clear from the first use of β in sentence 2.
– Yakk
Aug 23, 2021 at 15:07

Time dilation goes along with length contraction.

Consider a muon that we measure travelling at 50 km when it "should" have only travelled 600 m. We say it travelled 50 km because we have conventional reference frames that are convenient for us to use that say that distance is 50 km. The muon, however, is moving a substantial fraction of the speed of light relative to our reference frames, so lengths along its direction of travel are contracted in its own frame of reference (relative to ours).

In the muon's frame of reference, it only survived 2.2 microseconds, and travelled 600 m. In our frame of reference, both the time it survived and the distance it covered are different.1 The equations of relativity tell us how measurements of both time and length taken in different reference frames should correspond, and work out such that in no frame of reference (no matter how arbitrarily constructed) does measured distance divided by measured time exceed the speed of light.

When you ask "wouldn’t the bomb be able to tell that it has gone faster than light for the 2 microseconds it was alive?" you're mixing up frames. You're talking about the time measured in the muon/bomb's own reference frame but the distance measured in our reference frame (or at least, one that the bomb is moving 0.99 c relative to). If you measure the time and the distance in the bomb's frame its speed will be less than c, and if you measure the time and the distance in our frame the bomb's speed will be less than c.

1 The start and endpoint are the same in both frames though; it's the distance measured between those two points that is different, not whether or not the muon actually reached a given position.

• A good answer for the question thats really being asked..... Aug 23, 2021 at 9:20

In the frame of the bomb, you would travel away at 0.99 c for a distance of (0.99 c * 2.2 microseconds) before it exploded.

So, since in its frame you travelled away at 0.99 c from it, it would know, that in your frame, it travelled away at 0.99 c from you. In no frame, would it travel faster than c.

• Well using muons as an example, they should only be able to travel 600m if they live for 2.2 microseconds, yet due to relativity they can travel 10s of km before dying. Aug 21, 2021 at 17:22
• Yes, it would travel 10s of km IN THE LAB FRAME. But in it's own frame, the bomb would explode after 2.2 microseconds. In it's own frame, the lab and everyone else would travel .99c*2 microsecs , during the time it takes for it to explode. So, since your question was "wouldnt the bomb be able to tell........" , the answer is no, in it's frame it would survive for 2.2 microseconds
– user311898
Aug 21, 2021 at 17:26
• Alright, thank you! Aug 21, 2021 at 17:35
• @wanderer in the lab frame is its lifetime also longer? Aug 22, 2021 at 13:55
• @theonlygusti Yes, in the lab frame , it lives for longer due to time dilation
– user311898
Aug 22, 2021 at 14:12

if I put a bomb set for 2.2 microseconds and send it out at .99c, would it travel further than (.99c x 2.2 microseconds)?

It depends on which frame of reference you are considering. In the lab frame, it would indeed travel further than that. In its own frame, it would stay at rest while the lab would travel away to a distance of .99c x 2.2 microseconds before it explodes.

In the frame of the bomb, it stays alive for 2 microseconds, and within that duration of time, the earth, the lab etc. would travel away from it at .99c . So, it would not say that it has gone faster than light for the microseconds that it was alive.

• This is missing the distance "paradox" solution.
– Yakk
Aug 23, 2021 at 15:08

would it travel further than (.99c x 2.2 microseconds)?

If you send it out at a velocity, relative to Earth, of .99c, then Earth's spatial component (that is, the spatial component in Earth's reference frame) of the displacement between it being sent out and its exploding is more that .99c * 2.2 microseconds.

The bomb's spatial component of the displacement will be zero; in the bomb's reference frame, it is stationary. It will view Earth as moving in the opposite direction, and the distance it observes Earth moving will be .99c * 2.2 microseconds.

Now, that last statement is rather complicated, as the distance it observes Earth moving, by Earth's yardsticks, will be more than .99c * 2.2 microseconds. That is, if someone on Earth were to put out a ruler and put tick marks such that the tick marks are, in Earth's reference frame, each 1m apart, then the bomb will observe the ruler moving at .99c, and the tick mark next to the bomb when it explodes will be more than the number of meters that the calculation .99c * 2.2 microseconds gives (that is, .99c * 2.2 microseconds = 650 m, but the bomb will see more than 650 tick marks fly by). This is because in the bomb's reference frame, the ruler is traveling near the speed of light and thus contracted, and so the tick marks are less than 1m from each other.