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

  1. David says:

    Hello. Everything you all explain is clear to me and is correct…and when you think about it is pretty simple. But…time isn’t moving…light is moving and a clocks mechanism is moving at certain speeds and all that moves in opposed to each other reacts due to different speeds…but again no one is thinking of time as a stillness. Therefore:
    If I am starring at a clock which has one minute to 12 and I am travelling away from the clock at the speed of light, meaning that the image of the clock ticking one minute to 12 is moving towards my eyes the same speed as I am travelling, therefore it will allways be one to 12 for me but for other people who are standing by the clock,time will pass normally. So it is said that i will not age and everyone else will…..don’t think so.I will age the same as they do. If I counted seconds in my mind(not looking at the watch or my wrist watch)while travelling away,time would be the same.Why?If i didn’t travell into space,but rather around the clock at the speed of light, no one would see me,I wouldn’t see reality but an after image….but i could still count the same as the clock at the beginning of the story.My wrist watch may be off by years.But would age the same. A pilot in a jet counts time the same as you do, and it doesn’t matter how fast he goes. Clocks can trick you and make you think of time as something that moves.

    For example…I talk to a friend: Hey lets count 10 seconds together but I will make a trillion laps around the world at the speed of light during our count. Ok? And go!
    1,2,3,4,5,6,7,8,9,10.He would be 10 seconds older and so would I!

  2. David says:

    But than again….this is just my opinion;)

  3. anil says:

    David: I understood your example of “time” is “mechanical” movement of the clock and also time-space dilemma when you move away from the clock @ the speed of light but why time is slowing down with you travelling @ speed of light wearing that wrist watch?

  4. David says:

    Hello. Please read Vocation’s comments.
    Maybe it will be clearer to you than.
    For what it’s worth,I for one am 100 percent with his statement.
    You only see something…that doesn’t mean that it is what you see…understand?
    You are looking at the past…but it is not the past over there,is it? You are only deceaving
    yourself. And that is what I tried to say in my previous statement.
    I am stating that the movie “Interstellar” is wrong 😉

  5. Ifti says:

    I still don’t understand how alice would see the photon reach her at the same time as she was when she wasn’t moving sideways but Bob would definitely see a time difference

  6. Tom says:

    David: you have to trust a bit more in the people who actually learned A LOT about this. They are quite capable of understanding these kind of simple things. What you are presenting here is known as the Dunning-Kruger effect;

    What you are referring to in your post is only the socalled Doppler effect, and has nothing to do with the actual time dilation;

    Time dilation means all laws of physics are going at a slower rate, including how fast your mind works, so you won’t notice a thing. Altogether with that effect, you have the Doppler effect, which is what you described, which has no influence at all on what is actually happening, but is only about what is seen, which are two completely different things.

    There are only 2 constants at a base of special relativity;
    * a constant speed of light measured by every observer, regardless of their relative motion or of the motion of the source of the light
    * all laws of physics are the same for all observers in uniform motion relative to one another

  7. David says:

    Yes, I understand your point.
    But first of all:
    David says:
    September 19, 2015 at 9:25 am
    But than again…this is just my opinion;)

    After my comment, telling me that I am presenting the Dunning-Kruger effect, is just rude.
    How speed of light is held to be constant, when light going through Water travels with slower than light going through air, or when light travels through matters at ABSOLUTE ZERO temperature, travels at the speed of a bicycle ?What about longitudinal waves that appear to travel much faster than light? Or what about the proven mathematical correlation between particles that where once in contact but are now lightyears apart yet which under Bell’s Theorem seem to be instantaneously connected?
    Third: I know we are all talking about vacuum measuring,but you cannot use something effected by speed to measure time with. You can use something which is not effected by anything.
    Fourth:I know that GPS proves Einstain was right.(doesn’t prove that the expiration date of the satellite is wrong from Earths point.
    Fifth:Imagine I am in a shuttle and moving at…whatever…lets say half the speed of light.From a point, my clock is ticking faster than the speed of light? Just depending on how I am facing my clock(laser clock)I can change various outcomes of time measured?
    Six:You can’t prove I am wrong because you do not have a grandpa twin …you can prove time measuring differences on time measuring devices. Nothing else.

    The mistake that Einstein said was incorrect with relativity was that he did not take into account electricity and magnetism. This is now being included and the new learning has thrown relativity out the window.

    Any way….please make an experiment with two humans and show me. And than….if you are correct…..than you can be rude.
    Till than…you are relatively correct.

  8. The Physicist The Physicist says:

    Great questions!
    The important thing about the speed of light is the speed, not the light itself. There’s a post here that talks about that a bit. When light travels through a material is does a lot of starting and stopping; on average it travels slower, but when it moves it’s still moving at the usual speed (post here). Importantly, the speed of light through different materials has nothing at all to do with relativity.
    There are effects that seem to travel faster than light, but they never actually involve things doing so. A good example is this experiment. It sounds exciting, but even the folk who did that particular experiment are quick to say that they never actually got anything to go faster than light.
    Bell’s theorem implies some pretty wild stuff about how the universe works, but unfortunately it doesn’t involve anything going faster than light either. Every now and again someone will come along and propose a method of using entanglement to “cheat” relativity, but without exception there’s always been a flaw in their reasoning and their experiments never work (post here).
    Just a quick note, Einstein’s original paper was titled (in German) “On the Electrodynamics of Moving Bodies”. The theory of relativity was originally written to explain the relationship between electric and magnetic fields; EM is built into its bones.

  9. David says:

    Thank you very much for explaining, I understand.
    Quick thought: Lets say that we can go fast as near 300 000 km/h in a brand new rocket. And we launch the rocket(very slowly in comparison to the speed of light) and position our selves near the Moon from where we are going to engage in that high speed.
    Where is 0 km/h from where we will start? Does anyone know? Because we measure speed relative to Earth right?
    We already are on a vessel. Our galaxy, which is also on a vessel of kind.So we already are moving at high speeds and we would start at those speeds…so we basically wouldn’t have to reach 300 000,rather find a direction to which to go in order to maximize our speed relative to the stillness in space which is 0 km/h? Yes?

  10. David says:

    Or a better question? Is it possible to reach 0 km/h?

  11. Tom says:

    Hi David,

    Speed is relative, so everything can be seen as moving at 0 km/h, or ‘almost’ the speed of light. Light will always be seen as moving at it’s same steady speed, as observed from any of these observers, so light is actually a completely different realm than the realm of matter.

  12. Philip says:

    “you can prove time measuring differences on time measuring devices. Nothing else.”

    Actually it’s worse. You can only prove counting differences on counting devices. There are no devices that measure time. There are devices that count vibrations, oscillations or other periodically occurring physical events.

    Some of these physical phenomena are affected by motion. But there is no proof that any of those motions or differences in motion are at all related to some mysterious thing called time. Clocks count. They do not sample something a yield a reading.

  13. Philip says:

    Sorry for the typo. I meant to say at the end that clocks do not sample something AND yield reading.

  14. Steven McCarthy says:

    Einstein theory of relativity States that as you travel near the speed of light that time slows down and matter shrinks. The first time I saw this in modern physics, I thought it was only relative and did not really affect the individual in the accelerated time frame I started to rationalize how this could be true. In our low energy state the hydrogen atom has 100000 to 1 distance between the nucleus and the electron as you increase your speed, to the speed of light the distance between the nucleus and the electron decrease to 1 to 1 this cause everything to shrink. Also time will slow down and then eventually stop at the speed of light. The reason time stops is based on Einstein proven theory that speed of light is a constant no matter what reference frame you are in. Speed is equal to distance divide by time to keep the speed of light a constant when you distance shrink time must also decrease equally to to keep speed of light constant. To accelerate to the speed of light you must increase energy into object, by increase the energy into an object this will cause the electrons to move closer to the nucleus in the atom. This happens because the increase in energy overcomes the repulsive nuclear force, between the nucleus and the electrons. Note: This affect is the same as increasing the gravitational field around an object. The gravitational field over comes the nuclear force causing the electrons to move closer to the nucleus. This agrees with Einstein theory about the warpage of space time (space shrinking and time slowing down around massive objects)

  15. Tom says:

    That’s always the weird thing about time. It really only exists in particles interacting with the Higgs field: matter. Empty spacetime in essence holds no time: it’s just a mathematical means to understand how matter behaves within it. Photons have no time; they don’t ‘age’, and have no frame of reference.

    From our ‘matter’ point of view it almost seems they should have a frame of reference, but if our particles wouldn’t interact with (or far less well said; be slowed down by) the Higgs, then space and time couldn’t be derived out of the difference between these fields/particles.

    If we had just decided to use the lightsecond (299799458 m) as a distance unit, then c would have been equal to 1, and we wouldn’t have had to use it to translate E to M anymore in E=Mc^2 * lorentzfactor. Also it would’ve added to our intuition on how time and space are very much entwined.

  16. Tom says:

    Steven, once you reach inertial motion of an object, after some initial accelerating, there are no forces doing anything anymore, so you can’t really compare that to gravity, which is effectively similar to continuous acceleration. An object is ‘flattened’ in the direction it’s moving, relative to the observer, so to another observer, in another frame of reference, the same object is flat in a completely different direction.

    If it was simply the compression of a moving object by electrons moving closer to the nucleus, and time going slower, then all *other* objects from it’s own viewpoint would seem to be larger, with time going faster. But the thing about relativity is that both observers will obverse the other as being shrinked and going slower. So c acts like a sort of mirror, that way.

    This is why c is much more than just the speed of light: it’s a universal constant. It’s the connection of energy to mass, and space to time, and this is also why we need to think of our mathematical spacetime grid changing, depending on the frame of reference of the observer, to be able keep a correct helicopter view of all possible events within it. How this changes for different frames of reference, is all expressed with mathematical structures, known as tensors.

  17. bill says:

    this is for david,
    I’m bill, riverside ca.
    I understand the theory why time slows down at light speed, but does it really?
    I mean, five minutes is five minutes right?
    or would my wrist watch actually run slower if I were traveling at light speed and return to find everybody decades older?
    could you send me an email with your response?

  18. Urs says:

    Hi, I think both examples (the light clock and the rocket) do not explain or prove that time passes faster or slower depending on speed of an object. I think both proves are flawed. Let me explain:
    – the light clock: Your experiment starts off with a photon being bounced between absolutely parallel mirrors. The distance the photon travels is defined by the distance between the two mirrors. The direction of the photon is given by the mirrors it bounces off from. The mirrors being exactly parallel, the photon’s direction is exactly perpendicular to the mirrors. If I move the box with the mirrors while the photon is moving, the location where the photon hits the mirror the next time is different. The direction of the photon does not change only because I move the box. Eventually when I move the box horizontally enough the photon which is on its way from one mirror to the other will hit the wall of the box and the experiment is dead. For the experiment to keep working with the box in motion the mirrors would need to be slightly tilted in the direction of the motion, directing the photon to the location the center of the mirror will be when the photon hits that mirror. The distance of the photon travelled this way is longer because it now travels slightly at an angle. The duration the photon takes for that distance is exactly the same whether measured by the Bob or Alice. Your experiment wrongly assumes that the photon somehow magically changes direction because the box is moving, but it does not!

    – the rocket experiment is flawed too. Let me explain:
    First, let’s assume that both Bob and Alice send out the same type of white light from their sources of light (not to confuse the blue and red any further).
    We both agree that the light spreads at the same speed for both of them.
    Let’s assume it is a huge rocket that stands on the ground and light takes exactly one second to travel from Bob to Alice. It also takes one second to travel from Alice to Bob (there is no motion yet), the light is neither shifted to red or blue for either of them.
    Now, assume the rocket moves upwards at half the speed of light. A photon sent from Alice to Bob will now take 2/3 of a second (because the photon will travel at C and the movement of the rocket will shorten the distance by 1/3). Because Bob is now hit by higher frequency light, the light he preceives as blue shifted. The opposite is valid for Alice. She sort of runs away from Bob’s light source. A photon sent from Bob to her will take 1.5 seconds exactly (the time the light takes to travel from bottom to top of the rocket plus the distance the rocket travels in space during that one second). Alice is now hit by lower frequency light. She perceives the light as redshifted.

    While Bob and Alice see different wave lenghts and different colors their time or their clocks still show exactly the same lapse of time. In other words when our rocket moves at half the speed of light for 10 seconds, both Bob and Alice will have spent the exact same 10 seconds, one seing red the other seeing blue light. the fact that one sees higher frequency light and the other sees lower frequency light has no influence on how fast time has passed for them.

    Now, I realize I as a non physicist and non-mathematician stand alone with this argument amongst a world of geniuses and against a world of proves and experiments that confirm what you are stating is correct. Still, the arguments and so called proves you mention here do not make logical sense to me. I am very open to stand corrected and made understand

  19. Tom says:

    You seem to confuse inert speed with acceleration in both your arguments.

    – Einstein´s light clock
    Einstein´s clock is *inert* during the thought experiment that shows how time is affected by motion. So there´s no acceleration needed to show this. This results in a saw tooth type of motion of the photon, as seen from the observer that sees the light clock from a train that moves with constant speed. In other words: for the observer outside the train, the photon is moving sideways. The observer in the train disagrees, and will say it is only moving up and down. Same would go with a bumping ball within the train, for that matter…

    Once acceleration comes into play, you´re very right: there´s no reason for the photon between the mirrors to start moving sideways, if the train suddenly started to move. And it won´t. But this is not part of the thought experiment. Photons do have momentum, like the bumping ball, even though they don´t have a rest mass. So if there would be a laser light behind a top semitransparent mirror that points straight down to a parallel mirror at the bottom, and start shooting light downward, it won´t move straight down, as seen from the observer outside the train, if the train was moving sideways. This momentum originates from the energy within the photon (energy = mass).

    – rocket experiment
    Here, the rocket in Physicist´s example is accelerating, whereas it is moving at a constant speed in your given example.
    To follow your example with the rocket moving at a constant speed of 0.5c: first of all, there´s no length contraction of that amount with a speed of only 0.5c. At 0.6c there is a length contraction of 0.8, as seen from a static observer. Since Alice and Bob are no static observers, since they´re both on the rocket, this length contraction won´t be noticed, because of their time dilation. This time dilation will make it appear for both Alice and Bob that their lights are white, and not blue or red shifted at all, since they’re not accelerating. Frequency has a time element.
    As seen from a static observer, the light sent from the bottom arrives later at the top of the rocket that’s moving upwards at a constant speed, and light sent from the top arrives sooner at the bottom. But we also observe a time difference between the clocks of Alice and Bob, that is not there for Alice and Bob. This time difference exactly compensates for this.

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