# Special Relativity

Please read Introduction to Relativity first, or this may not make sense!

Special relativity explains the relationship between mass, time, space, energy, and the speed of light.

## Relative

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

• 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 600Â km/s

But in our daily lives we don't notice any of that.

We only notice relative speeds.

Next time you travel around, remember you are also moving relative to a vast array of cosmic objects, making our journey through the universe even more awe-inspiring!

### Example: you are in a moving train.

Have a game of table tennis!

The ball bounces back and forth just like you were at home.

If there were no windows, and the train was running on a smooth track there is actually no way you could tell how fast you were going in relation to the ground.

The laws of Physics are not affected by your speed.

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

## Acceleration

But we do notice acceleration!

So, to keep things relatively simple, we won't think about acceleration here.

Special Relativity does not deal with acceleration or gravity
(but General Relativity does).

## 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!

This was shown by the Michelsonâ€“Morley experiment in 1887 which found no difference in the speed of light "forward" or "backward" of Earth's orbit despite Earth moving quite fast through space.

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!

• 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.

## Frames of Reference

In that example we have two different frames of reference: you inside the box and Jade outside the box:

Your frame of reference is the box you are in. You are at rest inside the box, while the remaining universe, including Jade, is moving past you at high speed.

Jade's frame of reference is from outside the box, observing it moving towards them at high speed.

Both frames of reference are equally valid: there is no absolute frame of reference.

The laws of physics are the same for all observers, regardless of their frames of reference.

So you and Jade can describe the same events using your own frames of reference, and may observe different measurements of time, length, etc

## Inertial Frame of Reference

An "inertial" frame of reference is not accelerating and has no external forces acting on it such as gravity.

• objects in the frame have inertia: they keep a constant speed (which can be zero) unless acted upon by a force.
• the laws of physics are the same in all inertial frames of reference.

## A Moving Box

Let's do an experiment inside your 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 will see the light take a longer (slanted) path, BUT it still travels at the speed of light, so it must take more time for them, measured as r

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

## The Math!

Having accepted this so far, let us do some math! What are the distances?

• The distance for the inside observer is the speed of light by the time t: ct
• The distance for the outside observer is the speed of light by the time r: 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:

Move (vr)2 to left:(cr)2 − (vr)2 = (ct)2
(ab)2 = a2b2:c2r2 − v2r2 = c2t2
Simplify left:r2(c2−v2) = c2t2
Divide by c2−v2:r2 = c2t2c2−v2
Simplify right:r2 = t21−v2/c2
Square root:r = t 1√(1−v2/c2)

Which tells us how much time for Jade the outside observer (r) compared to you the 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)

That increase in time that an outside observer experiences is called "Time Dilation", dilation means getting larger.

It is really important when near light speed:

### Example: 99% of the speed of light in a vacuum

v = 0.99, c=1γ = 1√(1−(0.99)2/12)
Simplify:γ = 1√(1−(0.99)2)
Calculate:γ = 10.141067...
Calculate: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 ?

## Real Example: Muons

Muons* are elementary particles with a mean life of only 2.2 µs (2.2 microseconds). Light travels only 660 meters in that time.

So how do we detect muons here on the ground when they are created by cosmic rays colliding with nuclei in the upper atmosphere, many thousands of meters away?

How do the muons last so long?

Muons travel very close to the speed of light and get a large time dilation effect, with γ (gamma) about 9.

So for them 2.2 microseconds may have passed, but as outside observers we see about 20 microseconds passing, which is enough time for them to travel 6,000 m.

### Frames of Reference

Let's see what things look like from each frame of reference:

From Earth looking at a muon, we see it's clock ticking really slowly so it has enough time to reach the ground.

From the muon's point of view it lasts only a few microseconds, but it sees the distance from the sky to ground as really short!

The distance for the muon is short enough that it can get there in time!

Let's imagine a muon with a life of 5 microseconds.

And imagine it travels at 0.994c (γ = 9.1), which is about 298 × 106 m/s (note: the same in both reference frames).

In its own reference frame the muon would cover a distance of 1490 meters, but let's see it from both reference frames:

Earth's Frame of Reference:
• Muon zooming in at 0.994c
• Speed = 298 × 106 m/s
• Muon aging slowly
• Distance = 13600 m
• Time = 9.1 × 5×10-6 = 45.5×10-6 s
• Speed:
13600 m45.5 × 10-6 s
= 298 × 106 m/s
Muon's Frame of Reference:
• Earth zooming in at 0.994c
• Speed = 298 × 106 m/s
• Earth aging slowly
• Distance = 13600 m / 9.1 = 1490 m
• Time = 5×10-6 s
• Speed:
1490 m5×10-6 s
= 298 × 106 m/s

When γ increases the time in one reference frame, it must also reduce the length in the other reference frame.

Everything within a reference frame should make sense: we might see the muon lasting a long time, but within the muon's reference frame it lasts the normal time. The muon might see the Earth as squished in front of it, but it is perfectly normal in Earth's reference frame.

This "squishing" effect is called Length Contraction and happens only in the direction of motion:

Proper Length: the "non-squished" length of an object when it is not moving. It is the true length of the object, as measured in its own rest frame.
Proper Time (τ) is the time as measured by a clock that remains stationary relative to the events being observed.

The formula for length contraction is:

L = L0γ

Where:

• L = the observed length of the object
• L0 = the proper length of the object at rest
• γ = 1√(1−v2/c2)

## Summary So Far

• Gamma: γ = 1√(1−v2/c2)
• Time Dilation increases time by γ: t = γ t0
• Length Contraction reduces length by γ: L = L0γ

## Relativistic Speed

Relativistic Speed: a speed close enough to the speed of light that we should use relativity calculations. Everyday speeds such as cars and planes are "non-relativistic".

## Simultaneous?

Two events that are simultaneous (happen at the same time) in one frame of reference may not be simultaneous in another frame of reference

Imagine you are in a moving train and stand exactly in the middle of two clocks that are preset to flash a light at the same time:

You see the flashes arrive simultaneously
Jade sees the flash from one clock reach you long before the other clock

This concept challenges our simple ideas of simultaneity and shows that there is no universal "now" that applies to all observers in the universe.

## Adding and Subtracting Relative Speeds

So, how do we add speeds, 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.

When they are both heading in the same direction we can use this fomula:

vnew = v1 + v21 + v1v2/c2

### Example: A spaceship going at 0.6c launches a rocket (relative to it) at 0.5c, what does an outside observer see?

Let's use the formula above:

vnew = 0.6 + 0.51 + 0.6×0.5/12

vnew = 1.11 + 0.3

vnew = 0.846...

The two speeds combine to make 85% of the speed of light. Neat, huh?

### Example: Earlier we looked at this: What does happen when we add 0.9c to c?

Let's use the formula above:

vnew = 0.9 + 11 + 0.9×1/12

vnew = 1.91 + 0.9

vnew = 1

So the outside observer sees the combined speed of 0.9c and c as exactly c

So we never get above c

## One More Example

### Example: very very fast train

Let's look at what each person sees here:

0.6c = 180 × 106 m/s, and γ is:

v = 0.6, c=1γ = 1√(1−(0.6)2/12)
Calculate:γ = 10.8
Calculate:γ = 1.25

So Time Dilation = 1.25 and Length Contraction = 0.8

• Speed = 180 × 106 m/s
• Distance = 225 m
• Time = 1.25 × 1×10-6 = 1.25×10-6 s
• Speed:
225 m1.25 × 10-6 s
= 180 × 106 m/s
• Train incoming at 0.6c
• Speed = 180 × 106 m/s
• You (and train) age 1.25 slower
• You (and train) squished 80%
• Distance = 225 m × 0.8 = 180 m
• Time = 1×10-6 s
• Speed:
180 m1 × 10-6 s
= 180 × 106 m/s

For you:

## Mass-Energy Equivalence

In Special Relativity, mass and energy are interchangeable. This is shown by the famous equation:

E = mc2

Where:

• E = Energy
• m = mass
• c = the speed of light in a vacuum

This equation tells us that a small amount of mass can be transformed into a large amount of energy. The speed of light is a very large number, which means even a tiny amount of mass can release a huge amount of energy.

It is also possible to convert energy into mass, but you need a lot of energy.

## Real-World Applications of Special Relativity

Special Relativity has practical applications in science and technology, such as:

• GPS Navigation Systems: GPS relies on accurate time measurements. GPS satellites move fast, causing their clocks to experience time dilation. Special Relativity is used to ensure precise timing for accurate positioning.
• Particle Accelerators: Particle accelerators like the "Large Hadron Collider" use Special Relativity to work out relativistic effects during high-energy collisions.
• Nuclear Energy: Special Relativity explains the immense energy released in nuclear reactions. The equation E=mc2 is used as we convert mass into energy in nuclear reactors.
• Particle Physics and Cosmology: Special Relativity helps understand how matter and energy behave in extreme cosmic environments like suns, black holes and supernovae.

SN 1006 Supernova Remnant (NASA)

## Summary

• Gamma: γ = 1√(1−v2/c2)
• Time Dilation increases time by γ: t = γt0
• Length Contraction reduces length by γ: L = L0γ
• Adding speeds: vnew = v1 + v21 + v1v2/c2
• Mass-Energy Equivalence: E = mc2

Albert Einstein

## Footnotes

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 includes the effects of acceleration and gravity.

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

Light only travels that speed in a vacuum! It can travel slower, see Light

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

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.

### References:

Muons: https://web.mit.edu/lululiu/Public/pixx/not-pixx/muons.pdf

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