Time Dilation Vs. Length Contraction

[Kirsten-maths]

I'm having a major mental block to special relativity - i understand it the principle of it but seem to be struggling with time dilation and lenth contraction. I've spent a while thinking that time dilation and length contraction happen both togethor like you can't get one without the other... but my maths isn't working out, (and I'm a maths student...) is it either time dilation or length contraction? Or was i right the first time its both?

I'm using an example of a muon created in the Earth's atmosphere 3km up, with a speed of 0.99c

Cheers in advance for any help :)

 

[grav-universe]

Right, it is both. From the frame of Earth, the muon, if it had a radius, would be contracted. The muon's "clock" also time dilates at .99 c, so ticks slower, and the muon takes longer to decay as the Earth frame views it, so it passes further through our atmosphere. From the point of view of the frame of the muon, it is Earth that is contracted, so the atmosphere is less than 3 km as it enters it. The muon decays in its normal time within its own frame, so again, it travels further through the contracted atmosphere before it decays.

 

[yuiop]

I'm having a major mental block to special relativity - i understand it the principle of it but seem to be struggling with time dilation and lenth contraction. I've spent a while thinking that time dilation and length contraction happen both togethor like you can't get one without the other... but my maths isn't working out, (and I'm a maths student...) is it either time dilation or length contraction? Or was i right the first time its both?

I'm using an example of a muon created in the Earth's atmosphere 3km up, with a speed of 0.99c

Cheers in advance for any help :)

Usually there are 3 things you have to take into account. They are "length contraction", "time dilation" and "the relativity of simultaneity". The last item is usually overlooked or misunderstood and is usually at the root of most well known paradox examples. The equation for the relativity of simultaneity is given as $$\Delta T = L_0*v/c^2$$ where $$L_0$$ is the proper separation of two clocks at rest wrt each other in the rest frame of the clocks. $$\Delta T$$ is the apparent difference in the elapsed times of two clocks that are moving relative to an observer if the two clocks are synchronised in their rest frame. Even better, to avoid confusion when carrying out calculations, it is better to use the full Lorentz transformations as they automatically take the relativity of simultaneity into account, rather than the length contraction and time dilation equations .

 

[bcrowell]

The really simple, not at all rigorous argument is that c has to stay the same under a Lorentz transformation, and c is a distance divided by a time, so you can't just have length contraction without time dilation or vice versa.

 

[ghwellsjr]

The really simple, not at all rigorous argument is that c has to stay the same under a Lorentz transformation, and c is a distance divided by a time, so you can't just have length contraction without time dilation or vice versa.

Yes, and, like yuiop said, you also need the relativity of simultaneity. Without that, you'll never get c to remain c simply by dividing a contracted (smaller) length by a dilated (larger)time.

 

[bcrowell]

Ugh, two years later I've realized that my #4 was totally wrong. Length contraction and time dilation are not analogous. Here's a long thread about that: https://www.physicsforums.com/showthread.php?t=299857 One easy way to see that they're not analogous is that the world-line of a clock is a line, whereas the world-line of a ruler is a strip.