The force of gravity also causes time dilation to occur. In the aforementioned atomic-clock-jet-plane experiment, which was conducted on November 22, 1975 by a U.S. Navy P3C antisubmarine patrol plane, only about 10% of the time dilation experienced was due to velocity. In fact, a clock on the second floor of a building runs slower to an observer in a free-floating frame than one on the first floor. The equation for gravitational time dilation:

tf = to * √(1 - 2M/r)

Where:

tf = time for the observer in the free-float frame
to = time for the object undergoing time dilation
M = mass of the gravitational body expressed in meters (G/c2 * Masskg
r = the reduced circumference (C/2π)
The time dilation due to the Earth or even the Sun's gravity is quite small - the curvature of spacetime only differs from flat space by one part in a million in the Sun's case - but gravitational time dilation is quite measureable if there are any black holes nearby.

Time dilation due to a black hole's gravity can be determined by the simplified equation:

to = tf / √(1 - Ch/Co)

Where:

tf = time for the observer in the free-float frame
to = time for the object undergoing time dilation
Ch = circumference about the event horizon
Co = circumference of the orbiting object

Consider a black hole with a circumference of 10,000 kilometers. An hovering object with a orbital circumference of 20,000 kilometers will appear to have been orbiting for about 1.4 days for every day that passes in the life of the outside observer. Should that object drop to 10,001 kilometers, more than 100 days will pass in the free-float frame while only a day passes for the orbiter!

Sneaking even closer... should the circumferences differ by just a meter... spend just a day orbiting at that distance... and almost ten years will pass.

Perceptive folks will notice that when the object's circumference equals that of the event horizon, division by zero occurs. It's at that point that infinite time dilation occurs. I think that's why no one can observe an object crossing the event horizon, although I think it might have something to do with the infinite red shift of light at the event horizon. Maybe Pseudomancer has the answer...

Time Dilation is a physical effect that is a direct consequence of Einstein's Special Theory of Relativity. It is a result of the fact that the speed of light is the same in all reference frames. To see this, we will need a small amount of algebra and a little bit of ASCII art, but for the most part this is a conceptual argument.

First, ask yourself this: How do we tell time? How can we be sure we have a system for telling time which makes sense when converting from one reference frame to another? Since Einstein tells us that the speed of light is the same no matter what reference frame we are in, it seems like this would be a good candidate to ensure our system for measuring time was proper. Let's construct a clock, then, which measures time by determining how long it takes a beam of light can travel some fixed distance. The precision in our measurement of time will then be exactly equivalent to our precision in our measurement of distance. If our measurement of distance is invariant from one reference frame to another*, then our measurement of time should be accurate no matter what reference frame we are in.

So now we have a clock, which looks something like this:


  ====
   ||
   ||
   ||
   ||
   ||
   ||
   ||
  ====
What you're looking at is a pair of mirrors reflecting a beam of light up and down between them. For each one-way trip, the beam of light ticks off an amount of time equal to:

Δt0 = d/c

Where d is the distance between mirrors, and c is the speed of light.

Great clock, huh? Well, now let's see what this clock tells us when we look at it in a moving reference frame. If we put the clock on a train moving along the ground at a velocity v, the picture we see from the ground looks something like this:


        ====
         /\
        /  \
       /    \
      /      \
     /        \
    /          \  /
   /            \/
  ====         ====
The length of the trajectory has increased, and since the speed of light is invariant, the travel time has also increased! The length of time between events changes when in a moving reference frame. An experimenter on the train will measure the same Δt0 as before, but an observer on the ground will calculate a different Δt, which we can determine by using the pythagorean theorem:

(length of travel distance)2 = (height)2 + (amount it's moved in time t)2

(cΔt)2 = d2 +(vΔt)2

Δt2(c2-v2) = d2

Δt = d/√(c2-v2)

Now we use our equation above for the time Δt0 the observer on the train measures:

Δt = cΔt0/√(c2-v2) = Δt0/√(1-v2/c2)

Thus, the clock on the train ticks more slowly than one on the ground. The exact ratio has been calculated.

Immediately, a paradox presents itself. Einstein says that all inertial reference frames are "equally right"; there is no universal stationary frame. From the perspective of the observer on the train, the train is stationary while the guy on the ground is moving along with velocity equal to -v. Thus, the clock that the grounded observer is carrying should be ticking slowly from the perspective of the observer on the train! Each observer accurately calculates that the other's clock is ticking more slowly than his own clock! How is this possible?

The answer has to do with the relativity of simultaneity. Read that node if you haven't yet, because I'm not going to explain any more than I have to in one node. There are two main points that should be understood:

(a) Simultaneity of events is relative, depending on your reference frame, and

(b) The time-ordering of events can be relative, depending on your reference frame.

Given this, it should be no surprise that the question of whether two clocks "agree" or which is "faster" or "slower" should depend on our reference frame. To ask which clock reached 12:00 first is a question of the ordering of events in time, which we see is a relative question when the events are at different locations in space. "Comparing clocks" doesn't make any real sense unless the clocks are brought together to the same point in space. For further discussion of this issue, see twin paradox.

In addition to time dilation and the relativity of simultaneity, Special Relativity also predicts an effect known as length contraction.


*It isn't, but that doesn't matter for this argument, the reason being that the light-ray trajectory we are going to set up is perpendicular to the velocity of the other reference frame.

I'd like to expand upon the concepts of time dilation already covered in this node, and offer, what I think is, a more comprehensible proof of it that only involves the pythagorean theorem and the fact that c, the speed of light, is constant no matter the reference frame. I'll let the upvotes/downvotes be a judgement of the clarity of my explanation.

Time dilation is a result of Einstein's theory of special relativity, which was first published in 1905. It's a mathematical statement that relates the time elapsed in two different reference frames, moving at speeds relative to each other. The best-known example of time dilation is the famous twin paradox. Time dilation itself can be simplified to a trite statement like, "Time slows down when you move faster," but this is not quite true. Really, it shows how time is NOT an absolute thing, and time measured in one frame is not necessarily the same as time measured in another frame. It shatters the idea of simultaneity and can lead to many situations that, to us, seem paradoxical (see: the twin paradox).

Why does this occur? Essentially, it is a result of the constancy of the speed of light as measured from ANY reference frame. The speed of any particular photon, a beam/wave of light, they measure the exact same speed, no matter how much the difference in speeds between the two reference frames. This is odd, for one, because you would expect that the motion of light is similar to that of cars: you can perhaps move at a speed relative to the speed of light, and light will appear to "slow down," but this is not the case. The result of this? Time slows down in a moving reference frame. I'll show mathematically how this is so in a bit, and it only requires the constancy of c and knowledge of the pythagorean theorem to prove.

The standard illustration of time dilation involves trains. The reason for this is because during the times during which Einstein developed Special Relativity (the early 20th century), the the fastest conceivable mode of transportation was, in fact, a train. People commuted on them and they became the ultimate symbol of speed, being the fastest land transport around. This tradition has largely stuck among physicists, even though we have jets, airliners, and really fast cars.

Okay, here's the example: Dick is standing on a train platform. Jane is standing on a train which moves past the platform at a velocity v; for the sake of argument, assume v is large enough (greater than 10% the speed of light) for the effects of special relativity to be detected. There is a clock on the train that both Dick and Jane can see, as well as a clock on the platform. When the train was not moving, the clocks were calibrated and basically ticked perfectly in unison. However, When the train is moving a paradox occurs: Dick observes the clock on the platform and the clock on the train, and finds the clock on the train to be running slow (note that the clock on the train is moving WITH the train at speed v). When 2 seconds elapses for Dick, he sees less time elapsed elapsing for Jane simply by looking at the clock on the train. Jane observes the exact same phenomenon in reverse: because Dick is moving relative to herself, she see the clock on the platform to be slow while the clock with her on the train is perfectly fine. it's easy to see how this alone seems, to our feeble minds and our slow-paced lives, is a paradox. So who is right? Well, they both are, from their own reference frames. This discrepancy can be easily shown mathematically.

Okay. The set up for this proof is very similar to the one above. On the train are two mirrors, and a photon bounces back and forth between these two mirrors. Let t be time measured on Dick's clock for the light to go from the bottom top top mirror, and t' be time measured on Jane's watch for the same phenomenon, who is moving at velocity v relative to Dick. The mirrors on the train are oriented vertically and set a distance d apart. Here's what the experiment looks like to Jane, showing the path of the beam during one bounce:

-----
  ^
  |
  |
  |    d = c t'
  |
  |
-----

In time t' (measured on Jane's watch) the light beam travels a distance d = c t'. This is simply because distance = speed * time. Easy enough, right?

The next question is, What does Dick see? Well, consider this: the time Dick measures for the the light beam to hit the top mirror from the bottom mirror is t, which is presumably a longer amount of time than t'. However, he also notices that the mirrors themselves travel a distance v t horizontally. Thus, the light beam also has this extra distance to travel. To Dick, the light moves in a slanted path, forming the hypotenuse of a right triangle. The length of this hypotenuse is c t, the speed at which light travels * the time of its motion (according to Dick). Let's look at a diagram: (note my rather shitty attempt at a diagonal. Just assume it's straight)

-----            -----
                   ^
               ---/
              /    
  D= c t   ---    
          /
       ---
      /   
   ---
  /    x = v t      
-----  ------->  -----
Initial          Final
position      position
of                  of
mirror          mirror

Notice that the light still travels at the speed of light, only now it travels on a diagonal path. So Dick still measures c the same value as Jane would measure it on the train (he finds distance traveled, divides by time on his watch, and it remains c). One more thing to note: we can construct a right triangle from this. The vertical distance traveled by light remains the same even though the train is moving. We have a measure for this already: y = d = c t':


Diagonal line - path light travels according to Dick
vertical line - path light travels according to Jane

-----            -----
                   ^
               ---/|
              /    |
  D= c t   ---     |
          /        | y = c t'
       ---         |
      /            |
   ---             |
  /    x = v t     |
-----  ------->  -----
Initial          Final

That looks like a right triangle to me. And it is! So now we can find a relationship between t and t' using the pythagorean theorem.

(v t)2 + (c t')2 = (c t)2
c2t'2 = c2t2 - v2t2 = t2(c2 - v2)
t'2 = t2(c2 - v2)/c2 = t2(1 - v2/c2)
t' = t √(1 - v2/c2)
Or, as it is normally written:
t = t' * 1 / √(1 - v2/c2) = t'γ

Where γ is the Lorentz factor, equal to 1 / √(1 - v2/c2). This is given its own symbol because it appears so much in relativity.

So there you have it. A simple explanation of time dilation. Let's confirm the results: t should always be greater than t', except when v = 0. What happens as v approaches c? Well, the denominator of gamma becomes smaller and smaller, making gamma overall larger and larger. Thus, t increases as v increases, and is thus always greater than t' unless v = 0. So we got what we expected. Nifty, eh?

The implications of this idea are pretty amazing. Once scientists got used to the idea, it really made them question whether the idea of "simultaneity" is a valid one, and really had some interesting philosophical ideas. There is NO absolute reference frame, but the laws of physics hold true for ALL reference frames (that speed of light will always be c, &c.). There is no absolute marker for truth or true speed or anything, you can only designate things relative to each other. The fact that time isn't absolute was really an astounding idea.

But I digress. You can read about this in other articles, try special relativity. :)

Quod erat demonstrandum, biznatch.

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