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Recently I was watching Carl Sagan's Cosmos: A Personal Voyage. In episode 8 ("Journeys in Space and Time") there is a scene presenting the idea of time dilation, due to traveling close to the speed of light, here is an excerpt.

I understood it as follows: in order of the dilation to happen you have to travel close to or exactly at (which is presumably impossible) the speed of light.

Now, what I would like to know is:

  1. After which speed the time dilation starts to happen (I assume the closer to the speed of light the greater the dilation)?
    • How (if at all) this "tipping point" was (or could be) calculated?
    • Or does the dilation happen only when object reaches the exact speed of light?

What exactly puzzles me, is this: say there is you and a friend of yours, and you running around your friend in circles, with the speed of light (or fairly close to it), now as shown in the excerpt, time for your friend will go faster then for yourself (i.e. he will age greatly, while you won't).

Now, putting the speed of light aside, let's assume the same situation (i.e. you running around your friend in circles), but lets say not with the speed of light, but just "really fast". Say you are just moving 10 times faster than your friend, now it is obvious (at least it seems so to me), that time for your friend will go not faster, but slower (since you can do 10 times more things in a given time-span than your friend can).

Again, putting speed of light aside, if as shown in the excerpt, somebody left his friends and really fast went to some other place and returned, it is possible that they won't even notice he was gone. So my question is, basically, after which speed it stops being true and time dilation kicks in?

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  • $\begingroup$ In order to help you, it would be nice if you could mention what other resources you consulted before posting a question. Effects like time dilation have been very widely written about. It would be good to see in your question something like "I saw the picture of the light pulse clock on Wikipedia, and the formula, which seems to suggest that time dilation happens at any speed. Is that right?" $\endgroup$ – Andrew Steane Nov 18 '18 at 15:19
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There's no such speed 'limit', unless you count $0\ \rm m/s$. Even if you're moving really, really slow with respect to your friend, you'll measure a different elapsed time.

Look at the canonical formula, $$\Delta t'=\frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}$$ $\Delta t'$ is the time you measure between two events, $v$ is your velocity with respect to the other observer, and $\Delta t$ is the time measured by your friend. If $v=0$, then $\Delta t=\Delta t'$. But if $v$ is really close to zero (whether or not it is positive, because the direction of the velocity isn't relevant), you'll necessarily observe some minute difference, and you're only limited by the accuracy with which you can measure.

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There is no particular speed when time dilation kicks in, if I walk relative to you while you are stationary, time indeed moves solver for me albeit the difference is almost negligible but it is there. The formula you can use to calculate this time dilation yourself is this: $$T=T_0\sqrt{1-\frac{v^2}{c^2}}$$ Where suppose if you set $T_0=1s$ then $T$ can give you the value of how much less time passes for me(if I am moving at speed $v$ relative to you) while you are at rest and $1s$ passes for you.

Notice that if I was at a car moving at suppose $10 m/s$ (which is around $36 km/h$ or $22.4 mph$) then our times would be off by just $5.56\times 10^{-16}s$. So you can see why this is virtually impossible to notice in everyday life.

Now that doesn't mean that nothing in our lives is connected to this time dilation. A very important thing, that is GPS Navigation, requires time dilation to be involved or satellites can mistrack your position by almost $15m$ or about $50 ft$. As these satellites travel at around $4 km/s$, but still, this time dilation effect is tiny only in the orders of magnitude of $10^{-11}$.

Time dilation really starts to become more noticeable as you approach closer to the 90% mark of the speed of light, as even at 50% the speed of light, the discrepancy is only $0.133s$.

Here is a nice video on how time dilation allows us to detect short-lived muons, which without time-dilation, would be almost impossible to detect as they travel from the upper atmosphere to the surface: https://www.youtube.com/watch?v=rVzDP8SMhPo

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To get a time dilation of only 10%, say, you need to be going at $\frac{1}{\sqrt{1-v^2/c^2}}=1.1$, or $v\simeq .41c$, i.e. 41% the speed of light. That's 125,000 km/s!

What this means is that although, technically speaking, time dilation is present at all speeds, it only "kicks in" to the extent that a human would notice when you get to around half the speed of light. That doesn't seem like a big disadvantage, but actually this means that there will not be much length contraction below these speeds. Hence, if you want to travel and actually make it back home to your family, you are essentially limited by the speed of light. Without this, you could accelerate forever (given enough gas) and length-contract the universe to the size of a peanut.

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Time dilation is occurring at all speeds. It is the question of when it's extent is enough for us to be noticeable.

Now, according to general theory of relativity, both speed and acceleration (gravity) affect the phenomenon of time dilation.

Now, as far as your question related to friend running in circle is concerned, I would like to quote a similar example - the time dilation observed in artificial satellites.

Time dilation explains why two working clocks will report different times after different accelerations. For example, at the ISS time goes slower, lagging 0.007 seconds behind for every six months. For GPS satellites to work, they must adjust for similar bending of spacetime to coordinate with systems on Earth.

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