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Does Motion Affect The Speed At Which Time Passes? (Part 3)

Updated on August 17, 2016
The symmetry of special relativity may seem paradoxical at this point in our discussion, but we will tackle this issue in our upcoming articles
The symmetry of special relativity may seem paradoxical at this point in our discussion, but we will tackle this issue in our upcoming articles

Deriving The Relativistic Time Dilation Factor

Before we attempt to disentangle this next apparent paradox (i.e. which we were discussing near the end of Part 2), let us first have a go at deriving the formula for the relativistic time dilation factor (okay, let’s stop hyperventilating!), as it may shed more light upon the symmetry of special relativity. As our starting point, we retrieve the prior diagram from our thought experiment that shows the paths of each photon, as seen from our point of view, as they complete their respective “ticks”. So, here it is again, a bit more simplified this time.

The paths that the respective photons take, as observed from OUR point of view
The paths that the respective photons take, as observed from OUR point of view

We will therefore derive the time dilation factor using our vantage point as we stand at the train crossing, per the diagram. Because of the logic which we employed to lead us to the astonishing results of our though experiment, we now know that the phenomenon of time dilation is dependent on only one parameter – the relative speed, v, of the moving object. In our case, we consider ourselves to be at rest, and the moving object is Scotty’s train, that is, his train is moving at the speed v relative to us, and hence we will see his clocks ticking more slowly than ours, which are, of course, ticking at the normal rate of time passage. Thus, in keeping with this logic, we must be able to express the time dilation factor as a function of the relative speed alone. The strategy that we will employ for our derivation will be to calculate the amount of time, as measured by us using our watches, that is, as measured from our point of view, that it takes Scotty’s moving photon clock to complete one tick, given that it is moving at the speed v relative to us. We can then directly compare this amount of time to the (lesser) fixed, unchanging amount of time, again, as measured by our watches, that it takes our stationary photon clock to complete one tick. In doing so, we can then determine the quantity (i.e., the factor) by which the passage of time has slowed down for Scotty, as compared to the normal rate of time passage that we are experiencing. For instance, if we measure that it takes an amount of time, say, 5.25t, for Scotty’s moving photon clock to complete one tick, while we measure that it takes a (lesser) fixed amount of time of only, say, 1.50t, for our stationary photon clock to complete one tick, then we can directly conclude that: Time itself, for Scotty, is passing by at a rate that is 3.50 times slower than the normal rate of time passage, since 5.25t / 1.25t = 3.50t. And why can we make this deduction? Because, as the second conclusion of our thought experiment states, “…the rate at which Scotty’s photon clock (and, of course, all of his other clocks) tick, as observed by us, is obviously in direct proportion to the rate at which time itself is passing for Scotty on board his train.”. We will try to be as clear and concise as possible with our derivation. And so, hold on to your stomachs, because away we go!

STEP 1: Obtain an equation for the amount of time that it takes Scotty's moving photon clock to complete 1 tick

STEP 1: Obtain an equation for the amount of time that it takes Scotty's moving photon clock to complete 1 tick, as measured from OUR point of view
STEP 1: Obtain an equation for the amount of time that it takes Scotty's moving photon clock to complete 1 tick, as measured from OUR point of view

STEP 2: Obtain an equation for the diagonal distance d

STEP 2: Obtain an equation for the diagonal distance  d  in terms of the photon clock height  h, the relative speed of Scotty's train v, and the amount of time that it takes Scotty's moving photon clock to complete 1 tick
STEP 2: Obtain an equation for the diagonal distance d in terms of the photon clock height h, the relative speed of Scotty's train v, and the amount of time that it takes Scotty's moving photon clock to complete 1 tick

STEP 3: Solve for the amount of time that it takes Scotty's moving photon clock to complete 1 tick

STEP 3: Substitute this equation for d back into Eqn(1), and solve for the amount of time, as measured from OUR point of view, that it takes Scotty's moving photon clock to complete 1 tick, in terms of h, c, and v
STEP 3: Substitute this equation for d back into Eqn(1), and solve for the amount of time, as measured from OUR point of view, that it takes Scotty's moving photon clock to complete 1 tick, in terms of h, c, and v

STEP 4: Obtain an equation for the amount of time that it takes our stationary photon clock to complete 1 tick

STEP 4: Obtain an equation for the amount of time that it takes our stationary photon clock to complete 1 tick, as measured from OUR point of view, in terms of h and c
STEP 4: Obtain an equation for the amount of time that it takes our stationary photon clock to complete 1 tick, as measured from OUR point of view, in terms of h and c

STEP 5: Obtain the ratio that is the relativistic Time Dilation Factor

STEP 5: Compare the amount of time that it takes Scotty's moving photon clock to complete 1 tick [Eqn(2)] to the amount of time that it takes our stationary photon clock to complete 1 tick [Eqn(3)]; this ratio is the relativistic Time Dilation Factor
STEP 5: Compare the amount of time that it takes Scotty's moving photon clock to complete 1 tick [Eqn(2)] to the amount of time that it takes our stationary photon clock to complete 1 tick [Eqn(3)]; this ratio is the relativistic Time Dilation Factor
We used the fact that TRAVEL TIME = DISTANCE TRAVELED / SPEED OF TRAVEL and a little bit of high school algebra to obtain the equation for the relativistic Time Dilation Factor
We used the fact that TRAVEL TIME = DISTANCE TRAVELED / SPEED OF TRAVEL and a little bit of high school algebra to obtain the equation for the relativistic Time Dilation Factor

We have now obtained what we set out to derive – an equation for the relativistic time dilation factor, that is, the ratio of the rate at which time passes for Scotty as compared to the normal rate at which time passes for us, as observed from our point of view, in terms of only the relative speed of Scotty’s train v, and the speed of light c. But since we already know that the speed of light is a universal constant, then the only true variable in this equation is thus the relative speed v (how about that, eh?). Having said that little ‘brainteaser’, let’s next try to figure out what this equation actually means, and how it works!

“Enough With The Math – How About Some English?”

Let us now attempt to describe what this final equation is trying to say, using some English. We state once more that the relativistic time dilation factor is a ratio that has the following meaning:

with both amounts of time being measured by us using our (stationary) watches, where v is the speed of Scotty’s train relative to us, and c is the speed of light. Hence, this ratio is a direct measure of the rate at which time passes for Scotty as compared to the normal rate at which time passes for us, as observed from our point of view (we are not saying anything new here – just summing up and re-emphasizing what we have discussed and derived). We must now point out, however, that even though we used a “thought” experiment involving a fictitious train and fictitious clocks to obtain this formula, the formula itself is quite ‘real’, and, furthermore, it obviously does not just apply exclusively to Scotty’s train, but rather, it applies to any object, real or imaginary, that is moving at a speed v relative to a given observer.

To best illustrate the workings of the equation, we will examine a few cases where the relative speed v of Scotty’s train, or other objects previously cited, takes on particular values, specifically, those values which we have used throughout our lively discussion. The first case is when v = 0 (meters / second), that is, when Scotty’s train is at rest relative to us. In this case,

If Scotty's train is at rest relative to US, then the time dilation factor equals 1
If Scotty's train is at rest relative to US, then the time dilation factor equals 1

Therefore, the equation is saying that: “The rate at which time passes for Scotty is the same as the normal rate at which time passes for us, as observed from our point of view”. Well, dog – gone it, things had better be this way, as even common sense and intuition are correct in telling us that any and all objects that are at rest relative to us must be experiencing the same normal rate of time passage that we do! Putting it another way, if all objects that are at rest with respect to us had clocks strapped to them, then all of these clocks would, of course, be ticking at the same normal rate as our clocks.

The second case is when v = 0.866c, or 259,800,000 meters / second. If we remember, this is the speed, relative to us, at which Scotty’s train raced through the train crossing. We apply our newly discovered time dilation formula once again, and in this case,

If Scotty's train is moving at a speed of 0.866c relative to US, then the time dilation factor equals 2, as observed from OUR point of view
If Scotty's train is moving at a speed of 0.866c relative to US, then the time dilation factor equals 2, as observed from OUR point of view

Now, the equation is saying that: “From our point of view, it takes Scotty’s photon clock 2 times longer to complete one tick as compared to our photon clock. Therefore, the rate at which time passes for Scotty is 2 times slower than the normal rate at which time passes for us, as observed from our perspective.” And thus, all clocks on his moving train will tick at a rate that is 2 times slower than the normal rate at which our stationary clocks are ticking, and consequently, we will observe that for every second that elapses on our clocks, only ½ of a second elapses on Scotty’s clocks. Hence, as we have already found out, Scotty, along with everyone and everything else aboard his train, and indeed, the train itself, will age 2 times slower than we do!

The third and final case is when v = 0.998c, or 299,400,000 meters / second. Yet again, recall that this is the speed, relative to us, at which those darned muons blazed through our atmosphere on their way to a collision with the earth’s surface. Repeating our procedure, we apply the time dilation formula, and in this case,

If the muons are moving at a speed of 0.998c relative to US, then the time dilation factor equals 15.82, as observed from OUR point of view
If the muons are moving at a speed of 0.998c relative to US, then the time dilation factor equals 15.82, as observed from OUR point of view

This time, the equation is saying that: “The rate at which time passes for the muons is 15.82 times slower than the normal rate at which time passes for us, as observed from our point of view (on earth).” Thus, if we could strap clocks to the moving muons themselves, then these clocks would be ticking at a rate that is 15.82 times slower than the normal rate at which our stationary clocks are ticking, and as a result, we would observe that for every second that elapses on our clocks, only 0.063 seconds would elapse on the muons’ clocks! Therefore, as we already know, the muons will age 15.82 times slower than we do, allowing them to traverse the entire span of the atmosphere and smash into the earth!

So, as we can see, these cases also amount to a good demonstration of our previous conclusion that states, “the greater the relative speed, v, of a moving object, the slower time passes for the object”. However, to get a better ‘feel’ for how the time dilation factor changes (i.e., increases) as the relative speed of an object increases, let’s illustrate this relationship, that is, the

with a graph and a corresponding table of values.

Our graph shows that as the relative speed approaches the speed of light, the relativistic time dilation factor increases without bound; at the speed of light, the TDF has a value of infinity!
Our graph shows that as the relative speed approaches the speed of light, the relativistic time dilation factor increases without bound; at the speed of light, the TDF has a value of infinity!
A table of values for our graph
A table of values for our graph
By analyzing the behaviour of the relativistic time dilation factor, we come to the incredible conclusion that time stops flowing at the speed of light !
By analyzing the behaviour of the relativistic time dilation factor, we come to the incredible conclusion that time stops flowing at the speed of light !

Astronautics Anonymous

Now, looking at our graph and its corresponding table of values, we notice that the time dilation phenomenon, although ever present at all relative speeds (i.e., from greater than 0 to c), becomes noticeable only as the relative speed becomes comparable to the speed of light. For instance, if a spaceship is traveling at 90% of light speed relative to our space station, then from our point of view here on the station, we will observe that time on board the ship is passing 2.29 times more slowly than it is for us, while we, of course, experience time going by at the normal rate of time passage. And if the ship is traveling at 99.9% of light speed, then we will observe that time on board now passes 22.37 times more slowly. But if the ship’s relative speed increases only a little more, to, say, 99.999999% of c (which is only an additional 0.099999%), then we will observe that time on board now passes 7,071.07 times more slowly! Thus, we would also notice that for every 1.96 hours that elapse on our clocks here on the space station, only 1 second would elapse on the ship’s clocks! Indeed, as relative speed increases, the time dilation factor increases in greater and greater disproportion, and as we can see from our graph, the time dilation factor increases without bound as the relative speed gets closer and closer to the speed of light, c. This means that at the speed of light, the time dilation factor has a value of infinity, and so, we would observe that time on board the spaceship was passing an infinite number of times more slowly than it was for us! And therefore, we come to the incredible conclusion that time stops at the speed of light! Hence, if the spaceship could travel at the speed c[1], and if we could somehow “look” into the ship, then we would observe a ‘frozen image’ of the astronauts and their clocks inside! We can illustrate this mind – blowing result using the equation for the time dilation factor. Recalling that the

However, from the astronauts’ point of view, just as we’ve emphasized for Scotty’s train a number of times now, they and their spaceship are at rest while it is we and our space station who are moving past them at the various relative velocities discussed above. Thus, the astronauts will experience the normal rate of time passage, and they will observe that it is our clocks that are ticking 2.29 times more slowly as the space station moves past at 90% of light speed, 22.37 times more slowly as the space station passes at 99.9% of c, and 7,071.07 times more slowly as the space station passes at 99.999999% of c! And, from the perspective of the astronauts, if our space station could travel at the speed of light[1], and if they could somehow “see” what was happening on the space station, then they would observe a ‘frozen image’ of everyone, every clock, and everything else inside the station!

Indeed, the symmetry of special relativity, in particular, the symmetry of the second postulate - i.e., the constancy of the speed of light - MUST result in the astronauts deriving the same equation for the time dilation factor as we have derived !
Indeed, the symmetry of special relativity, in particular, the symmetry of the second postulate - i.e., the constancy of the speed of light - MUST result in the astronauts deriving the same equation for the time dilation factor as we have derived !

Astronauts Aren’t Just Space Junkies – They’re Good At Math And Physics, Too!

Certainly, this symmetry in observations between the two sets of observers, or as physics refers to them, the two frames of reference (i.e., the space station and the spaceship), is a direct result of special relativity’s two fundamental postulates, which we have discussed in detail. Indeed, if the astronauts were to derive the equation for the time dilation factor from their point of view or frame of reference, that is, their spaceship, (which to them, is at rest), as it applied to the clocks on the moving space station, then they would obtain the same equation which we had derived previously (in our thought experiment) from our point of view, specifically, that the

Here, v is the speed of the moving space station relative to their spaceship, and c is, of course, the constant, unchanging value of the speed of light. And why should they arrive at this same equation? Well, continuing to look at things from the astronauts’ perspective, if the astronauts had their own photon clock on board, then they would observe that its photon travels in the vertical direction only, moving up and then down to complete 1 tick. And they would therefore measure, using their watches, that the amount of time that it takes their photon to complete this tick, is equal to

where h is the distance between the reflective surfaces of the mirrors. On the other hand, they would observe, just as we had observed on Scotty’s train, that the photon in our space station’s photon clock travels the same double diagonal path to complete 1 tick, due to the space station’s relative motion. And since the astronauts would also measure the speed of light to always have the constant value c (as per the second postulate), then they would therefore also calculate, again using their watches, that the amount of time that it takes our photon to complete this tick, is equal to

which is of course greater than the amount of time it takes their photon to complete 1 tick. Hence, upon comparison of these two amounts of time that the respective photon clocks take to complete their ticks, the astronauts would come to conclude that our photon clock, and consequently, time itself, are running at a slower rate than the normal rate of time passage that they are experiencing, by a factor of

which is the time dilation factor! And since the length contraction phenomenon is a direct result of time dilation, then length contraction along the line of relative motion must also be symmetrical between the two frames of reference. Hence, just as was discovered with the muons, the astronauts would observe that the space station’s length (along the line of relative motion) is shorter than its normal length, also by a factor of

while we would conclude that the spaceship’s length is shorter than its normal length by the same factor.

Notes:

[1] In reality, no material object, that is, anything that has mass, can travel at or faster than the speed of light. The reason for this is rooted in Einstein’s famous equation,

E = mc2,

which equates mass with energy, essentially stating that mass and energy are two forms of the same thing! We will be discussing the implications of this equation in detail, including its imposition of the ‘cosmic speed limit’, c, but for now, we quote Einstein and L. Infeld once again, from their book “The Evolution of Physics” (1938).

The velocity of light forms the upper limit of velocities for all material bodies… The simple mechanical law of adding and subtracting velocities is no longer valid or, more precisely, is only approximately valid for small velocities, but not for those near the velocity of light.”

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