Introduction to Relativity

First, a little story:

Alex and Sam are two immortal super-beings

(...some time later...)

Now let's learn about our (real) Universe!

Relative

There is no "central reference point" in the Universe. The universe is expanding in all directions and everything is moving apart from everything.

So we can only measure speed relative to us or something else.

Example: when we say a car is going 100 km/h (about 60 mph) we mean relative to the ground

But think about this:

• The Earth spins on its axis at 1670 km/h at the equator
• The Earth orbits the Sun at 30 km/s
• The Sun moves around the Galaxy at 220 km/s
• Our Galaxy is moving relative to local galaxies at about 600 km/s

But we don't notice any of that.

We only notice relative speeds.

(Imagine how weird it would be if light, electricity, magnetism behaved differently at different speeds!)

Speed of Light

Light travels at almost 300,000,000 meters per second (to be exact: 299,792,458 meters per second) in a vacuum. That speed is called c.

c = speed of light in a vacuum

And c is the same in all directions!

The Michelson–Morley experiment in 1887 tried to find any difference in the speed of light "forward" or "backward" of Earth's orbit but found no difference. Despite Earth moving quite fast through space.

It was a surprise at the time.

So c is constant and is not affected by any relative speed.

c is the same for all observers, independent of the motion of the source!

Imagine you zoom towards your friend Jade at 0.9c and shine a light ahead of you:

• You will see the light moving at c
• Jade will also see the light moving at c (not 1.9c)

You might think we should add c and 0.9c together to get 1.9c, but the Universe does not work like that. Many, many experiments have proven that.

(We are talking speed here, no acceleration. Technically it is called an "inertial frame".)

A Moving Box

To learn how this works, imagine you are inside a moving box.

You measure the time it takes light to reach a detector, and get the answer t

Jade is outside and sees the box move past at speed v.

Jade sees the light take a longer (slanted) path, BUT it still travels at the speed of light, so it must take more time, measured as r

• t is the time for you, the inside observer
• r is the time for Jade, the outside observer

That is the key point. Having c the same for everyone means that time can be different.

Having accepted this, let us do some mathematics! What are the distances?

• The distance for the inside observer is the time t by the speed of light: ct
• The distance for the outside observer is the time r by the speed of light: cr
• And the outside observer sees the box move this far: vr

So these are the distances:

Which we can put into one diagram like this:

It is a right-angled triangle that we can solve using Pythagoras formula:

So now we can calculate how much time for an outside observer (r) compared to an inside observer (t).

That last term is so important it gets called gamma (the Greek letter γ) or "the Lorentz factor":

γ = 1√(1−v²/c²)

Gamma Examples

Here are some γ values for different speeds (you can do the calculations yourself, but will need the Full Precision Calculator):

Remember that c is about 300,000,000 m/s

Highway speed of 100 km/h (28 m/s): γ = 1.000 000 000 000 004 29...

There are 14 zeros in there, so almost exactly 1. That is why we never notice these time effects at the speeds we normally go at.

Jet speed of 2,000 km/h (556 m/s): γ = 1.000 000 000 001 72...

So even at jet speed there are 11 zeros in there.

GPS satellite orbit speed of 14,000 km/h (≈4000 m/s): γ = 1.000 000 000 084 1...

Still almost exactly 1. A clock on a satellite would be affected by about:

0.0000000000841×24×60×60 = 0.000 007 seconds every day

That is 7000 nanoseconds, but GPS needs about 20 nanoseconds accuracy, so we actually need relativity calculations to make GPS work. (Note: there is also a gravity effect, but we are not looking at that here)

10% of c: γ = 1.005...

Even going at 10% of c (about 100 million km/h) there is only a .005 difference

50% of c: γ = 1.155...

90% of c: γ = 2.294...

99% of c: γ = 7.089...

At 99% of c, for every day inside the box someone outside experiences a week.

In the style of the triangle we saw earlier it looks like this:

You can check for yourself: does 1² = 0.99² + 0.141² ?

99.9% of c: γ = 22.4...

Adding and Subtracting Relative Velocities

So, how do we add velocities, then?

For speeds that we normally experience it is OK to just add them, you won't notice anything wrong.

But for very fast speeds we need to think of relativity.

In the case of velocities heading in the same direction we can use this formula:

v_new = v₁ + v₂1 + v₁v₂/c²

Mysterious?

This may seem strange, but not really mysterious. It is just a direct result of the speed of light in a vacuum being a constant for all observers.

The strangeness is likely because in our everyday world we never have to deal with it. Maybe if the speed of light was only 100 km/h we would have a better "feel" for it?

Albert Einstein

Footnotes

That was an introduction into "Special Relativity" which does not include any effects of acceleration or gravity.

Albert Einstein released his special theory of relativity in 1905, but it took until 1915 for him to release his general theory of relativity that does include the effects of gravity.

Note also that when we mention the speed of light, or c, please remember:

And even though it is called the speed of light, it applies to the whole Electromagnetic Spectrum, Gravity Waves and more (basically any particle without mass).

And remember: we all experience the same laws of physics.

Faster than c?

Space can expand faster than c, but as far as we know no wave/particle can go faster than c.