At some point, we’ve all heard about time dilation – every sci-fi fan among us in particular. And yet, moving clocks DO slow down. This is not a fiction fantasy. It’s a little thing called Special Relativity.

We like to think of time as a fundamental variable that flows freely and uniformly, independently from everything else, and in a single direction.

Yet there you have it! From the onset… how can there be such a thing as a fundamental variable? It sounds like an oxymoron.

Reality is not what it seems… Time itself turns out to be a paradox. Time passes more slowly in places, more rapidly in others.

You cannot disentangle time from space. And as it happens, matter warps both time and space.

What’s so Special about Relativity?

Special relativity is based on the idea that all observers in uniform motion must agree about the laws of physics.

For example, all observers must find that light travels at a constant speed of

$c = 2.998 \times 10^8 m s^{-1}$ .

When the laws of electromagnetism are included – what with light being an electromagnetic wave and all that – it leads to nothing short of a revolution in ideas about the nature of space and time.

Suddenly, both concepts can be wrapped together into the single concept of spacetime.

And different observers in different states of uniform motion may disagree about which events are simultaneous in spacetime.

In fact, Einstein’s physics announced that:

- Moving clocks run slow.
- Moving rods contract.
- Simultaneity is relative.

Above all,

- The speed of light in a vacuum is a fundamental speed limit.

Moving Clocks: A Thought Experiment

Bern’s clock tower inspired Einstein’s famous thought experiment.

In 1905, Albert Einstein predicted in one of his famous thought experiments that assuming two clocks are brought together and synchronised, and then one clock is moved away and brought back to its starting point, then the clock which had undergone travel would now be lagging behind the clock which had stayed put.

In 1911, he restated and elaborated on this result:

“If we placed a living organism in a box […] one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations. For the moving organism, the lengthy time of the journey was a mere instant, provided the motion took place with approximately the speed of light.”

Albert Einstein

Einstein considered this to be a natural consequence of Special Relativity. Not a paradox.

But why?

Frames of Reference

In Physics, a frame of reference is essential for measuring things.

Indeed the description of a body’s motion depends on the frame of reference from which the motion is observed.

Two observational frames of reference. The second frame with origin O’ is moving away with respect to the first one with origin O, at a constant speed v. Diagram: Bartleby

We are talking about a coordinate system, with

- a point of origin O,
- orientated axes x, y, z, and
- a scale, specified by a set of reference points (x, y, z).

A mathematical viewpoint.

Reference frames are useful to specify the relationship between a moving observer and the phenomenon under observation.

This observational frame of reference implies that the observer is at rest with respect to the frame, even though he or she may not be located at its origin.

There are two types of observational reference frames:

1) Inertial Frames

A cartoon showing a ginger and white cat fast asleep on a small plank held balanced by a couple of billiard balls just before the balls get poked by a billiard cue. Image: NaturPhilosophie with AI — A cat remains at rest unless…

An inertial frame of reference is one where all laws of Physics take on their simplest form.

Frames of reference in which Newton’s First Law of Motion holds true are called inertial frames or rest frames.

According to the Principle of Inertia, a body remains at rest or in a state of uniform motion unless it is acted upon by an unbalanced force.

2) Non-Inertial Frames

Contrastingly, a non-inertial frame of reference is one in which fictitious forces must be invoked, in addition to Newton’s Second Law of Motion, to explain the observations.

Fictitious forces include such ones as the Coriolis force and the centrifugal force.

In an inertial frame of reference, a black ball moves in a straight line. However, the observer who is standing in a rotating (i.e. non-inertial) frame of reference sees the object following a curved path due to the Coriolis or centrifugal forces present in this frame.

In flat spacetime, the use of non-inertial frames can be avoided.

Measurements with respect to non-inertial reference frames can be transformed to an inertial frame by incorporating directly the acceleration of the non-inertial frame as that acceleration as seen from the inertial frame.

The Lorentz Transformation

For this, we use a set of very simple equations, well-known to undergraduate students, and often called the Lorentz transformation equations:

$t' = \gamma (t - \frac{vx}{c^2}) \\* x' = \gamma (x-vt) \\* y' = y \\* z' = z$

where $(t, x, y, z)$ and $(t', x', y', z')$ are the coordinates of an event in two frames with their origins coinciding at $t = t' = 0$ , where the primed frame is seen from the unprimed frame as moving with speed $v$ along the $x$ -axis and

where $c$ is the speed of light, and the factor $\gamma = (\sqrt{1 - \frac {v^2}{c^2}})^{-1}$ is the Lorentz factor.

So that’s the technical background to this answer.

Because we need to consider this problem from different observers’ points of view, that is to say different frames of reference.

Einstein’s Light Clock

Here we look at the Poincaré-Einstein’s Light Clock.

The majority of clocks measure how many times a repetitive action is carried out.

In the case of a digital watch, the quartz crystal usually vibrates 32,768 times a second. Electronic circuits count these vibrations. After 32,768 “ticks” have been counted, one second is added to the watch’s display.

Light Clocks at Rest

Now, the light clock is made up of two reflective mirrors placed at point A and point B. To make it work, we need to bounce a pulse of light between the two mirrors that are a known distance apart.

Again, the light $c$ travels at a speed close to 300,000 km per second (or 186,300 miles per second).

We have this pulse of light that goes up and down, up and down…

Assuming we separate the mirrors by a distance of 150,000 km (or 93,150 miles) i.e. half the distance 300,000 km (or 186,300 miles), each mirror will be struck by the pulse of light once a second. And the round trip from mirror 1 to mirror 2 and back again will also take the light pulse one second. Right?

In other words, we got ourselves a clock.

The very same principle can be applied to radar technology whereby electromagnetic signals are beamed out and reflected at very close to the speed of light, allowing for the distance to surrounding objects to be determined.

The light is restricted to a linear path from the bottom mirror to the top mirror, and vice versa.

Nothing extraordinary here.

Just a stationary clock. No big deal. Simples.

Light Clocks in Motion

Actually, there isn’t that much special about a non-stationary clock either… as long as we – the observers – are on the same moving platform as the light clock is – the same inertial frame. (Observer 1)

Things begin to get weird when we imagine that the moving clock is also being watched by an external and stationary observer. (Observer 2)

Two diagrams depicting Einstein's Light Clock at rest and in uniform motion. The left diagram represents the path of light seen by an inertial observer at rest respect to the light clock. On the right, an observer at rest is looking at the clock moving from left to right at a constant speed. — Einstein’s Light Clock Source: Researchgate

To understand further what is happening with Special Relativity, we need to board a rocket and you are flying with us…

We, the rest frame observer, whiz past the external observer while holding the clock to the porthole in the rocket.

Will both observers see the light clock doing the same thing?

The resounding answer is “No!”

When Things Get Weird…

To us, savvy science-minded individuals on board our light clock-equipped rocket, the light pulses just go up and down, up and down the same way we would expect them.

However, to the external observer, the light follows a very different trajectory, one that could be made out of a series of triangles.

Using Pythagoras’ Theorem, we obtain the length of the hypothenuse $h$ of those triangles:

$h^2 = o^2 + a^2 \\* = (150,000 km)^2 + (75,000 km)^2 \\* = (22,500,000,000 + 5,625,000,000) km^2 \\* = 28,125,000,000 km^2 \\* \\* \therefore \\* \\* h = \sqrt (28,125,000,000 km^2) \\* = 167,705 km$

And we multiply this result by 2 to obtain the return path of the light in 1 second:

$h \times 2 = 335,410 km$

What The… (Bleep)!!!

That result is confusing to say the least. Not perhaps what we had expected.

“Damn right impossible!!” you realize by now. ” Nothing goes faster than the speed of light. Not even light!”

So, what’s happening here?

Although the Maths are correct, the result appears wrong. Even Albert Einstein could not believe his own equations..

Observer 2 knows the distance a pulse of light can travel in a single second cannot be more than 300,000 kilometres, because he also knows that the speed of light is a constant.

If the speed of light cannot change, can anything else change?

Einstein reflected about this problem and came to an astonishing conclusion:

If the speed of light remains constant, it must be spacetime that changes.

Einstein realised that what Observer 2 would really see would be a light clock that apppeared to slow down. The “ticks” of the clock would now appear to be slower as viewed by Observer 2 than as viewed by the inertial frame Observer 1 on the rocket.

Experiments have been performed that support Einstein’s theoretical prediction of time dilation.

The rocket moves at 50% of the speed of light. Hence, according to Observer 2, the time the light pulse takes to get to the top mirror and back again would be about 1.1 seconds. The rocket, and everything on it, would be running in slow motion.

A Theoretical Demonstration

Now, there are a number of practical problems with such a clock.

The most obvious one is the separation distance of the mirrors, but we could put them closer together and just count faster “ticks”.

The large separation demonstrates the principle. It makes the Maths easier.

In reality, the mirrors would absorb some of the light everytime they got struck by the light pulse. And after a time, the light pulse would dissipate entirely.

Also, the fact that we can see the light at all means that at least some of it gets scattered, further weakening the signal.

However, none of this matters. Here we are dealing with a theoretical proof – not an experimental one.

Having said that…

Moving Clocks: A Practical Experiment

An accurate clock at rest with respect to one observer may be measured to tick at a different rate when compared to a second observer’s own equally accurate clock.

This effect arises neither from technical aspects of the clocks nor from the fact that signals need time to propagate, but from the very nature of spacetime itself.

So time dilation is an actual difference of elapsed time between two events as measured by observers either moving relative to each other, or differently situated with respect to a gravitational mass.

Onto General Relativity…

General Relativity arised from a desire to express physical laws in the same way for all observers, even for those who are not in uniform motion.

Ultimately, the motion of bodies is determined by the curvature of spacetime caused by the gravitational fields of other more massive objects.

Einstein’s General Relativity became a theory of gravity.

But as far as Special Relativity and time are concerned, we must realize that it all depends on that ‘proverbial’ observer.

There is NO absolute time in the Universe. It is as Einstein conceptualized it. Simultaneity of time is always relative…

NaturPhilosophie

Why Moving Clocks Do Slow Down

What’s so Special about Relativity?

Moving Clocks: A Thought Experiment