Q: Why does going fast or being lower make time slow down?

Physicist: Back in the day, Galileo came up with the “Galilean Equivalence Principle” (GEP) which states that all the laws of physics work exactly the same, regardless of how fast you’re moving, or indeed whether or not you’re moving.  (Acceleration is a different story.  Acceleration screws everything up.)  What Einstein did was to tenaciously hold onto the GEP, regardless of what common sense and everyone around told him.  It turns out that the speed of light can be derived from a study of physical laws.  But if physics is the same for everybody, then the speed of light (hereafter “C”) must be the same for everybody.  The new principle, that the laws of physics are independent of velocity and that C is the same for everybody, is called the Einstein Equivalence Principle (EEP).

Moving faster makes time slow down: I’ve found that the best way to understand this is to actually do the calculation, then sit back and think about it.  Now, if a relativistic argument doesn’t hinge on the invariance of C, then it isn’t relativistic.  So ask yourself “What do the speed of light and time have to do with each other?”.  A good way to explore the connection is a “light clock”.  A light clock is a pair of mirrors, a fixed distance d apart, that bounce a photon back and forth and *clicks* at every bounce.  What follows is essentially the exact thought experiment that Einstein proposed to derive how time is affected by movement.

The proper time "τ" is how long it takes for the clock to tick if you're moving with it. The world time "t" is the time it takes for the clock to tick if you're moving with a relative velocity of V.

Let’s say Alice is holding a light clock, and Bob is watching her run by, while holding it, with speed V.  Alice is standing still (according to Alice), and the time, \tau, between ticks is easy to figure out: it’s just \tau = \frac{d}{C}.  From Bob’s perspective the photon in the clock doesn’t just travel up and down, it must also travel sideways, to keep up with Alice.  The additional sideways motion means that the photon has to cover a greater distance, and since it travels at a fixed speed (EEP y’all!) it must take more time.  The exact amount of time can be figured out by thinking about the distances involved.  Mix in a pinch of Pythagoras and Boom!: the time between ticks for Bob.  So Bob sees Alice’s clock ticking slower than Alice does.  You can easily reverse this experiment (just give Bob a clock), and you’ll see that Alice sees Bob’s clock running slow in exactly the same way.

It turns out that the really useful quantity here is the ratio: \frac{t}{\tau} = \frac{C}{d} \frac{d}{\sqrt{C^2 - V^2}} = \frac{C}{\sqrt{C^2 - V^2}} = \sqrt{\frac{C^2}{C^2-V^2}} = \sqrt{\frac{1}{1-\frac{V^2}{C^2}}} = \frac{1}{\sqrt{1-\frac{V^2}{C^2}}}.  This equation is called “gamma”.  It’s so important in relativity I’ll say it again: \gamma = \frac{1}{\sqrt{1-\frac{V^2}{C^2}}}.

It may seem at first glance that the different measurements are an illusion of some kind, like things in the distance looking smaller and slower, but unfortunately that’s not the case.  For Alice the light definitely travels a shorter distance, and the clock ticks faster.  For Bob the light really does travel a greater distance, and the clock ticks slower.  If you’re wondering why there’s no paradox, or want more details, then find yourself a book on relativity.  There are plenty.  Or look up Lorentz boosts.  (The very short answer is that position is also important.)

The lower the slower: Less commonly known, is that the lower you are in a gravity well, the slower time passes.  So someone on a mountain will age (very, very slightly) faster than someone in a valley.  This falls into the realm of general relativity, and the derivation is substantially more difficult.  Einstein crapped out special relativity in a few months, but it took him another 10 years to get general relativity figured out.  Here’s a good way to picture why (but not quite derive how) acceleration causes nearby points to experience time differently:

Redder light at the top, bluer light at the bottom.

Alice and Bob (again) are sitting at opposite ends of an accelerating rocket (that is to say; the rocket is on, so they’re speeding up).  Alice is sitting at the Apex (top) of the rocket and she’s shining a red light toward Bob at the Bottom of the rocket.  It takes some time (not much) for the light to get from the Apex of the rocket to the Bottom.  In that time Bob has had a chance to speed up a little, so by the time the light gets to him it will be a little bit blue-shifted.  Again, Alice sees red light at the Apex and Bob sees blue light at the Bottom.

Counting the blue crests is faster than counting the red crests. However, since it's all the same light beam the number of crests has to be the same to everybody.

The time between wave crests for Bob are short, the time between wave crests for Alice are long.  Say for example that the blueshift increases the frequency by a factor of two, and Alice counts 10 crests per second.  Then Bob will count 20 crests per second (No new crests are being added in between the top and the bottom of the rocket).  Therefore, 2 seconds of Alice’s time happens in 1 second of Bob’s time.  Alice is moving through time faster.

Einstein’s insight (a way bigger jump than the EEP) was that gravitational acceleration and inertial acceleration are one and the same.  So the acceleration that pushes you down in a rocket does all the same things that the acceleration due to gravity does.  There’s no way to tell if the rocket is on and you’re flying through space, or if the rocket is off and you’re still on the launch pad.

It’s worth mentioning that the first time you read this it should be very difficult to understand.  Relativity = mind bending.

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19 Responses to Q: Why does going fast or being lower make time slow down?

  1. Pooja says:

    Thank you for the answer.Though I can’t claim to have fully understood it(given my limited knowledge in physics),the idea is clearer in my head than before.It’s a very intellectually overwhelming concept.If you ponder over it for a while,there is a sort of restlessness and discomfort which begins to creep in your mind.

  2. The Physicist Physicist says:

    I couldn’t sympathize more.

  3. Pingback: Q: According to relativity, two moving observers always see the other moving through time slower. Isn’t that a contradiction? Doesn’t one have to be faster? « Ask a Mathematician / Ask a Physicist

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  7. Rani Sunarso says:

    I still have question about twin paradox.

    Let’s say Alice is holding a light clock in a rocket, leaving earth, going up from the North Pole with speed of 0.9 c, while Bob watching her on earth. After a few hours, Alice comes back to earth and meets a very-very old Bob. Why Bob is getting older faster than Alice, while according to Alice, she didn’t move at all and it was Bob (and the earth) who moved down, leaving her rocket with speed of 0.9 c.

  8. Rani Sunarso says:

    Sorry, wrong place to ask. I will ask in the twin paradox page.

  9. The Physicist The Physicist says:

    No worries!
    In a nutshell, the croockedness of Alice’s path means that she experiences less time. It’s not immediately obvious why, but the twin paradox page does try to cover it.

  10. Vishwas says:

    In the example you gave by using the Pythagoras theoream ( light clock) , would the result be the same if you would have taken an another human running at a normal speed ( not with speed of light ) ? In other words, what is the significance of using some object having the “fastest speed” known ( ie. photon as we know till now). Why can’t i use some bit larger speed say 100m/s etc.

  11. The Physicist The Physicist says:

    It’s not the fact that light is the fastest speed that’s important, it’s that it’s always the same speed that’s important. From every point of view the speed of light is the same, whether you’re moving or not, while the “speed of runner” changes depending on how you move.

  12. Pingback: Q: How can wormholes be used for time travel? | Ask a Mathematician / Ask a Physicist

  13. Time dilation is so important (and REAL) that SatNavs have to make a 0.45 billionths of a second time dilation adjustment every day or else, after a week, your reported position would be 10 kilometers in error. (See my blog post http://bit.ly/wXBr9i ” A Subjective View of Time”)

    Now here’s a mathematical absurdity that comes out of velocity-related time dilation! If I move my hand 0.5m away from my body at 0.5m/s then it moves into a different reality than the reast of my body because of the time dilation effect.

    This of course presupposes that reality unfolds one Planck frame at a time (5.39 x 10^-44 s).

    What’s more, every part of the moved arm will be in a different time zone (at the planck scale) relative to every other part. Ludicrous? Seemingly so. That suggests to me that time is either an illusion (a conscious construct) or there’s something VERY wierd about its relation to perceived reality!!

  14. Lennie says:

    I’m still a little jogged off by this whole paradox-time-space-continium of a thing. If i could get a clearer view of what this all adds up to. And i’m starting to doubt that Flash is actually fast, what if evryone else is just super slow ? Einstein! Your a mind blower!!

  15. rigney says:

    We can probably lay every physics problem at Einstein’s doorstep, but the on I like best is him using a mirror to explain acceleration and C. Going from 0 to C. @ light speed would still still takes one full second. Looking in the mirror as you accelerate, the question is: Woud your image gradually fade as you approach C? And would your image disappear completely when you reached C.? So, to me time must be relative if it takes a photon traveling @ C., 13+ billion years to reach us from deep space? I hope this makes some sense.

  16. Pingback: Q: If a photon doesn’t experience time, then how can it travel? | Ask a Mathematician / Ask a Physicist

  17. Vocation says:

    I don’t really understand how time could slow down for someone. In that experiment with Alice pointing a laser down on Bob and that Bob was seeing blue light. Wasn’t Bob just seeing blue light because he was seeing the laser at C + the acceleration of the rocket? So he was seeing the light faster.

    If I where to stop time and move at all I should be blinded or at least my vision should be blurred because there is no true image reaching my eyes. How is light = time… Also if I did stop time or slowed down time I should be dead if I moved; the friction I created would or should have burned me to ashes.

    Its like seeing a plant 4 light years away. We SEE/PERCEIVE it as 4 years in the past while in reality its just as old as we are or as old as the universe.

    So how does the relatively of light = time? I do not understand…..

  18. Aditya says:

    From you article, I could make that it is the property of the clock and not ‘time’ it self which makes time slow down ? I don’t completely agree with the explanation.

  19. Xerenarcy says:

    we can understand almost all time dilation by examining what happens to light’s frequency under different circumstances. this works because distance and time as properties of space are quantities related to one another with the absolute constant c – the speed of light…

    light (photons) has an energy associated with its frequency. this energy is a conserved constant value for each photon – a specific photon will not change its energy, it carries that energy in the electromagnetic field as a distinct quantity. because energy is proportional to frequency, and frequency multiplied by wavelength is proportional to velocity (L * f = c), photons have the convenient property that if their wavelength changes since emission, it must have been due to relativistic motion of the observer or the warping of spacetime (either one, perhaps both) on the path to the observer that altered the wavelength.

    so then wavelength of light is a convenient ‘stick’ to measure changes in ‘spacetime density’ or curvature, because it directly affects the apparent color (frequency) of light passing through it (blueshift for compressed space, redshift for stretched space).

    in a gravitational field, mass compacts spacetime closer together towards the center of gravity… if you were to aim a green laser on a heavy planet from one point on the surface to another, it wouldn’t change color because the gravitational field (curvature of spacetime) hasn’t changed. aiming the laser into space, it will appear redshifted, because spacetime is less compacted away from a source of gravity, therefore the wavelength of light will be stretched out (arguably the light has lost energy by escaping the gravitational field). if you did the reverse, shining the same laser onto this planet from space, since spacetime becomes more compact towards the planet, the laser will appear blueshifted (arguably light has gained energy / ‘accelerated’ by falling into the gravitational field).

    how does this translate to time? quite easily. frequency is simply the inverse of period, or duration. if frequency appears to increase, but no physical quantity has changed (energy is conserved), it follows that the time in which we have made this observation of frequency, has been stretched out across more cycles.

    to make this relation between time and frequency more clear… for example, if we count 5 cycles over 5 seconds, we would conclude a frequency of 1 per second. if we met a person after the fact who told us: “that wave you were looking at, it was actually 0.5 cycles per second when i sent it down to your planet”, what then? well, 0.5 cycles suggests we would need 10 seconds to observe 5 cycles… we did it in 5 seconds, and we know that the number of cycles per distance traveled didn’t change and velocity can’t change, so those 5 seconds of our experience must have been stretched out across 10 seconds from the point of view of the sender.

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