Everything moves slower when going faster. And it has nothing to do with time!

Imagine you are the captain of a next generation spaceship, approved for top speeds of 20% of the speed of light. Other than that, the ship is huge and even has a baseball court, where you demonstrate your special pitching abilities by throwing a baseball always at the exact same speed of $50m/s$.^[1]And of course it goes in a straight line, since we are in space, where gravity only visits on occasion.

When the interviews and cermonies are done, you take your ship for its maiden voyage and promptly burst to its maximum speed, eager to show off and throw the first baseball on your new ship. Of course, our glorious throw will be live streamed to earth for everyone to watch. At the exact moment you pass by earth again, you throw the baseball. It's your most accurate yet, as your speed gauge proudly displays.

But when you come home, your friends looks tell you something is wrong. They've watched the stream. And they've measured the ball's speed, like millions of other people. And they have found your perfect track record for same speed throws now has a slight scratch. You threw it too slow. But your own measurements on the ship told you it was the perfect $50m/s$ throw. What has happened?

Reality's trolling you

Sooner or later, Einstein's theory of relativity ruins everybody's day. Your instruments didn't lie, they measured correctly, but they were affected by your ships speed as much as you and your baseball.

The one thing we really need to understand is that energy and mass are essentially the same thing. When you put energy into some kind of energy storage, like a battery, it actually gets heavier as well. It's an undetectably small difference, but it's still there.

Any kind of mass can store kinetic energy. When you throw a marble, you put energy into it, and it will keep moving, until all this energy is used up by friction or impact. Yes, a moving marble too gets heavier, but again the difference is undetectable.

To give some perspective:

The energy a nuclear plant produces in a year, weighs about as much as one litre of air. On the other hand, a grain of rice contains the energy produced by nuclear bombs used in WW2.^[2]Our current bombs produce the equivalent of 1000 grains of rice.

Now there is the wide spread misconception out there that matter and mass are the same thing. They are not! Matter is the phenomenon of being able to touch something, which has nothing to do with mass. Waves have mass, light has mass, but both are hardly touchable. So things can get heavier, without "putting stuff into it".

Speed and mass

Just for a moment, imagine we have a magical battery, where we can put in any amount of energy we like. When it's empty, it is just a normal battery and quite easy to throw. Now we charge it. Allot. In fact, we charge it until it weighs double its empty weight. Now throw it again! Did you now throw it faster or slower?

Let's say we take our ship again and accelerate to 20% the speed of light. By the way, the speed of light is usually written as $c$, which I will do from now because I'm lazy. At 20% $c$, we put a considerable amount of energy into our ship. Since everything on the ship is flying at the same speed, everything gained some of that energy, depending on its own weight. This means everything is heavier now! Not just your baseball, but your own arm, your air and you too!

Comparing objectivity

Hang on just a moment. You measured the baseballs speed on your ship. Why didn't it show that you threw slower because of the higher weight?

Because there is no such thing as an objective measurement. Every measurement ever performed was a comparison between something and the measuring instrument. Measuring length compares the length of your scale with the length you want to measure. Measuring time with your quartz watch compares time to the duration of one swinging in the quartz crystal.

Our highly modern instrument to measure speed compares speed to the speed of its electrons.^[3]It is more complicated than that. There are photones involved, being reflected from a moving object, but it boils down to exactly that. These electrons, which move inside of your ship, are just as fast and therefore are also heavier. So their speed is slower, and by the exact same proportion as your baseballs speed. So your measurement is absolutely correct. It's just not what other absolutely correct measurements would reveal.

Now time is measured by counting repetitions. How often the sun went up, or how often the moon was full, or how often a pendulum went back and forth.

Every clock in the world works exactly like a pendulum clock, we just found smaller and more precise pendulums. Your wristwatch probably uses a quartz crystal, very precise atomic clocks actually still use electrons in a crystal, but use caesium atoms to check if they keep their speed of repition constant.

So we need some process which repeats itself and takes a precise timespan. But this timespan will take longer, if the repeating object is charged with more energy, since it will be heavier and therefore slower!

The simple fact that everything becomes slower by the exact same proportion results in an effective slowing of time itself.

Messing with your head

So why does the theory of relativity state that time slows down and not that weight increases?

Because it is easier to calculate. Imagine you have a world map which slowly ages from the early earth with one huge continent to its modern state with several continents. Now when we slow this movie down in some places, it is a lot easier to imagine those places put in hony, slowing them down depending on the amount of hony around. Try imagening instead of hony everything getting heavier. What is easier for you?

And mathematics agrees. We treat time as a dimension like lengths, so we can treat time and lenghts as equal. This little trick would not be possible with mass. Mass does not have a direction like space and time, therefore behaves completely different.

But there is nothing special about time slowing down at high speeds, it is all just due to the fact that energy and mass are the exact same thing. Mass never becomes energy and energy never becomes mass, they are the same. Mass however is not matter. Matter is simply something that can be touched, it tells us nothing about its internal energy and mass!

Problems

We've found out that everything gets slower at higher speed since it gets heavier. But there is no reason why everything should get slower by the same amount. It could be that very heavy things get a lot slower than light things or the other way round! How could we conclusively show that everything slows down equally?

We just use some good old Backyard-Maths!

Prepareing to calculate

Let's just compare a flying ball while standing and while in motion. It is a very slow flying ball, so its own speed won't store much energy and won't contribute much to its mass, so we will ignore it.

Conservation of knock

A ball gaining weight will slow down thanks to the law of conservation of knock. The knock you feel when hitting a ping-pong ball depends on the ball's mass and velocity, if it is perfectly reflected. If I throw the ball faster, you will feel a harder knock, equally if I put some weight inside the ball. And our universe has this peculiar quirk to never ever let good knock go to waste. Knock has to be conserved at all times. Usually you will pass on the knock of the ball to the paddle through your arm to your whole body to your legs and through your shoes to the ground. But since it is such a light knock, you won't even notice.

Say the spaceship is standing still and the ball leaves the ship because John left the airlock open^[4]... again!, giving its knock to whatever it hits.^[5]Let's all hope it's John!")

Besides the spaceship we put a huge iron plate facing the spaceship, reflecting the ball back to the airlock. It's a smooth plate which can't slide^[6]magic!, but can be knocked. Furthermore it is very, very long.

Because our second experiment is to let our ball fly out of the airlock again, with exactly the same velocity, but this time while the ship is moving. The plate has to be knocked exactly the same! Otherwise knock would not be conserved.

Gather the ingredients to make some time sorbet

Since we use backyard maths, there won't be any real numbers or fancy things like that. We just have our knock while standing and our knock while flying at high speeds.

We actually borrowed our knock from physics. There, knock is defined as mass times velocity and called “momentum”, but that sounds too fancy.

To make real good backyard maths, we need some shortcuts:

• We will call the mass of the ball $m$ (for mass),
• its velocity $v_B$ (for velocity of the ball),
• the ship's velocity $v_S$ (for velocity of the ship),
• its kinetic energy $E$ (for energy),
• and the speed of light $c$ (for cpeed).
• The ball's knock we will call $p$, because all of the fancy physics books do that and we want to show them that we are just as good!
• Furthermore when we talk about the moving ship, we will mark all values with a tiny $n$ (for new).

Our knock is the same as momentum in physics: $p=m\ast <!--\; \ast \; -->v$

Kinetic energy has something to do with mass: $E=\frac{m\ast <!--\; \ast \; -->v^2}{2}$

Now finally for some real shady backyard-maths!Kids: Don't try this at home!")

Our plate always has to be knocked exactly the same, therefore: $$p={}_{n}p$$$$m\ast <!--\; \ast \; -->v_B=m_n\ast <!--\; \ast \; -->v_{Bn}$$

So our knock now equals the knock at high energy with a higher mass. The new mass is just the old mass plus whatever mass results of the kinetic energy. Our ball flies very slowly, but our ship is flying at 20% the speed of light. For our backyard calculations we can just ignore the ball's own speed and calculate its energy just by the ship's speed:

Gaining weight

Now we insert $m\ast <!--\; \ast \; -->c^2$ for Energy to calculate the additional mass $m_{add}$:

$$m_{add}\ast <!--\; \ast \; -->c^2=\frac{m\ast <!--\; \ast \; -->v_S^2}{2}$$ ... and divide by $c^2$:

Our ball's mass while flying is composed by its initial mass and the mass it gained from its kinetic energy:

again, we insert for $m_{add}$:

and extract $m_{add}$:

Knock conservation

Now we calculate the knock of $m_n$ with the equation from the very beginning:

We divide by $m$:

And divide by $v_{Bn}$ for good measure:

And there you have it. The right part tells us exactly how much faster our ball is when the ship is standing, and it is completely independent of the objects mass, so everything gets slower by the exact same proportion!

But is it correct?

Well, it is an approximation. We said our ship managed to go 20% $c$. That is $c\ast <!--\; \ast \; -->0.2$. let's replace our $v_S$ by $c\ast <!--\; \ast \; -->0.2$:

The $c^2$ part annihilates, but the $\frac{{0.2}^2}{2}$ stays:

Now what would relativity say to that?

At 20% $c$ it would yield a factor of 1.02062 ... for $\frac{v_B}{v_{Bn}}$

So we are off by less than a thousandth! That's very good for backyard calculations.

To recap: We just took the conservation of knock and the fact that mass and energy are exactly the same and found out that things on a fast moving ship will move slower independently of their mass. The only reason why we didn't hit the relativistic relation perfectly is because of our kinetic energy. This is actually just an approximation for the actual kinetic energy, but it is usually exact enough for mechanical problems.

To whom it may concern

... that we never even touched the theory of relativity:

Is this really a valid explanation of time dialation?

Yes, absolutely. In relativity this factor we calculated is called the Lorentz factor, usually named $\mathrm{\gamma <!--\; \gamma \; -->}$ and used to describe kinetic energy (among other uses):

Divide by $c^2$:

The new mass is again composed by normal mass and extra mass:

Again with the momentum:

divide by $m$ and $v_{Bn}$:

Now we assume that our ball has to pass a set length $s$ on the standing and the moving ship. We measure the time the ball takes to pass:

insert:

which is the basic equation to describe time dialation. So yes, in the theory of (special) relativity, slowing time or increasing mass are exactly equivalent.