So, how can this work in practice?
Look at the two images below.
The symbol \Delta
in \Delta t
denotes an interval of time. In general \Delta
means difference between the final value of something and its initial value.
On the left, I am sending a pulse of light up to a mirror and it bounces back. The time it took for the trip back and forth is just the distance divided by the speed
time = \frac{distance}{speed}
On the top figure the observer is at rest and the time measured is
\Delta t = 2L/c
This is just distance divided by the speed, in this case the speed of light.
Now another observer passes by and he/she measures the time on his/her clock (denoted \Delta t'
pronounced "delta t prime") between the start and the end of the pulse of light.
using "prime" is just a notation to differentiate between two cases. Here \Delta t
is the time interval measured by the observer at rest. \Delta t'
is the time interval measure by the observer moving
As you can see from the figure 6, the distance travelled as seen by the observer moving is bigger that as seen from the observer at rest with respect to the mirrors. We have that 2D>2L
, the factor of 2 is because the light goes back and forth. But the speed is supposed to be the same. Since \Delta t' = \frac{\rm{distance}}{c}
and the distance is bigger, it must be that the time is longer
\Delta t' > \Delta t
This is time dilation
. If I measure the time between two events on my watch, and you measure the time between the same two events but you are moving with respect to me, the time you measure will be longer than mine.
- S: Ok that's pretty cool. Do you have a real life example? It's hard to grasp.
- M: Not many, the speed has to be very high (with respect to c) for this effect to be noticeable. If an astronaut moves at 11 km/s around the Earth for one year, then his/her clock will be slow (it will be behind) ours by 0.02 s (after one year!). It is really a small effect. However, it becomes really important for satellite communication, and in particular Global Positioning Systems(GPS). GPS satellites are moving relative to the earth, and relative to the car/airplane that is using it. Without special relativity corrections, GPS could start to be off by as much as 10 kilometers per day, and get worse as time goes by!
- S: Wow, so small, but sometimes the effects can add up.
- M: Yes, it is important, since this is how the notion of a speed limit makes sense. The faster you go, the more time delay you get. It also affects the measurement of length it turns out. In the opposite way. Length gets contracted.
- S: All of this because light needs to travel at the same speed for all observers moving at constant velocity.
- M: Yes, measurements in spacetime adjust themselves in such a way that the speed of light is constant. It's a property of the spacetime tapestry we live in.
Time dilation and special relativity can be confusing. The thing to remember is the person that moves is the one who ages less compared to the one at rest. To remember this, you have to take the right point of view. For example, imagine that an astronaut is spending 80 days in space at high speed. We look from Earth and see him/her moving and we measure that the astronaut took 81 days to do all the stuff he/she was doing (we measure a longer time because the astronaut was moving with respect to us). The exact number 81 days can be calculated and it depends on the speed of the astronaut, this is just an example with really high speed. When the astronaut comes back, he/she has aged by 80 days while people on Earth aged by 81 days. The clock of the moving astronaut ran slow. Everybody felt time flowing normally. But the different observers did not age "in sync".