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]]>- Our current interpretation of time has issues (specifically we can’t find evidence of the Ether that physicists proposed)
- The two postulates below are valid:
- The laws of physics are the same in all inertial frames of reference.
- The speed of light in free space has the same value
*c*in all inertial frames of references

Our derivation involves the use of a light clock which is simply a beam of light bouncing between two ‘perfect’ mirrors. We will define a ‘tick’ of the clock as the light bouncing up and then down again.

Now imagine two light clocks in these two frames:

- A stationary frame (your frame, left image) and;
- A moving frame (a train moving right past you, right image)

In the stationary frame we observe the light going straight up and down. However in the moving frame as the light bounces between the mirrors the mirrors move some distance so the path taken by the light is at an angle (like a triangle). If you can understand the paths traveled by the light in these two frames then the hard part is over. The rest is simply trigonometry and algebra.

(1)

speed equals distance over time.

The proof consists of:

- Establishing a relationship between the distance between mirrors and the time it takes for a single ‘tick’ in each frame
- Combining the relations
- Simplifying and rearranging terms

Rearrange equation 1 for distance.

(2)

Now subbing in values (distance between A and B) and (speed of light) in the stationary frame the light travels a distance of

(3)

per every ‘tick’

In the Moving frame we can see that the lights path is at an angle so establishing a relationship with the time elapsed and distance between the mirrors involves using Pythagoras’ theorem

(4)

From the green triangle in the picture.

(5)

This is the distance the mirror moves in half a tick where:

- velocity of the moving frame and;
- is the time it takes to complete a tick in the moving frame

Note: and are used to distinguish how much time has passed in stationary frame and moving frames respectively.

is the distance between A and B.

(6)

We also know what is, it is the distance the light moves in half a tick which is:

(7)

Where c (lowercase) is the speed of light.

putting everything together:

(8)

(9)

At this stage we will transpose the equation so it is in terms of (this is purely manipulating the equation above)

Rearrange for

(10)

Factor out

(11)

Rearrange for

(12)

(13)

(14)

we can substitute equation (3), (our stationary frame equation) in to get

(15)

(16)

(17)

(18)

Typically physicists define

(19)

this gives us

(20)

So what does this equation mean? well remember earlier when i said and are different, well I assumed that. If I was wrong then the math would simplify to

(21)

which would mean time intervals are the same in all frames. This is the old Newtonian representation of time (moving like a arrow) that we discussed earlier. The new Einstein representation of time is very counter-intuitive, it implies that time behaves differently; instead of being fixed it depends on relative velocities between observers.

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Friends:So how is uni going?

Me:Its going great, I was learning about time travel the other day.

Friends:Time travel? like in the movies.

Me:No, not exactly, but theoretically its possible to travel many years into the future and hardly age at all.

Friends:…

Me:But it’s true, we have known about this for many years.

Start of every future conversation:Hey Eric, How’s time travel going?

Me:…

The point I’m trying to make here is that time travel sounds ridiculous and I don’t blame my friends for that, (if someone had told me this and I didn’t have a physics background I would be skeptical too).

My objective of this article isn’t to completely convince you of this phenomenon but to cast doubt in your mind into how you currently perceive the flow of time.

Intuitively, time passes the same for everyone whether you live in Melbourne or Greenwich, having fun or bored, are moving or staying still. Sir Isaac Newton’s was a famous supporter of this interpretation of time.

Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external …

Sir Isaac Newton, Newton’s Principia. https://en.wikisource.org/wiki/Page:Newton%27s_Principia_(1846).djvu/83

Here time is thought of as an arrow moving exactly the same for all observers. This made complete sense to physicists at the time considering there really wasn’t any evidence to say the universe behaved otherwise. To justify this theory, physicists introduced the concept of the ether. The ether was a medium which permeated the universe and was considered an absolute frame of reference.

How fast are you moving right now? you may be sitting down at a table or couch not moving at all, you may otherwise be in a car, train, bus, plane or some other vehicle. In any case it really doesn’t matter since the earth is moving **really fast** with respect to the sun, approximately 30 km/s (9 times faster than one of the fastest fighter jets in the world NASA X-43).

Now here is when it gets interesting, the Sun is moving approximately 19.4 km/s with respect to the black hole in the center of the milky way galaxy and our galaxy is moving approximately 580 km/s with respect to other galaxies. Now, how fast are the galaxies moving and what do we measure that with respect to. The point I wish to make is that **we need to measure our speed with respect to something else. **The ether is our way out of this conundrum, it is an absolute frame, one of which (it if exists) we can measure our speed against.

We now move to 1887 where Albert A. Michelson and Edward W. Morley proposed to establish evidence for the Ether. Since they knew the Earth revolves around the Sun they assumed that light would travel at different speeds depending on its direction. Kind of like if you were to throw a ball from a moving car, if it is thrown in the direction of travel the velocities add, and if thrown in the opposite direction the velocities subtract.

Here we have a light source (in red) that is split into two with the semi-silvered mirror (think of this as one of those one way mirrors at police stations). Both beams of light will then strike their respective mirrors, join back at the semi-silvered mirror and go into the detector. Physicists expected the light from the beams to be out of sync since for one of the paths (green or orange) would receive an additional speed boost.

Surprisingly, no matter how this device was rotated both beams of light were always in sync. Of course one experiment is not enough to completely refute the Ether theory but many other experiments were conducted and could not support the theory. Essentially this experiment gives evidence that **Light always travels at the same speed.**

Before I can mathematically derive this (which will be done in a future post) we must come to an agreement on two claims:

- The laws of physics are the same in all inertial frames of reference.
- The speed of light in free space has the same value
*c*in all inertial frames of references

source: https://en.wikipedia.org/wiki/Postulates_of_special_relativity (yeah I know its Wikipedia, but surprisingly Wikipedia isn’t a terrible source for Physics or Computer science)

Hopefully the Michelson Morley experiment (above) has convinced you of the second postulate, so now we can focus on the first.

Whether you are stationary or moving at a constant speed the laws of physics are the same. If you don’t believe me this try this experiment: while not moving throw a ball in the air, as expected in will come back down. Try this again in a moving frame e.g. (a car or train that is moving at constant velocity) the same thing will happen (don’t throw the ball outside of the car otherwise the wind will change its trajectory). The ball won’t move backwards it will follow the same path. This is because even though you think you are moving it is just as valid to say that you are stationary and the whole universe is moving. After all when you threw that ball in the air the earth was orbiting around the sun and the sun around a super massive black hole.

My proposed experiment doesn’t exactly prove that the laws of physics are the same whether you are moving or not. However the fact we are moving through the universe very fast and everything feels ‘normal’ should suffice as evidence.

If we believe the two postulates mentioned above, then our current perception of time falls apart.

Observe the picture above, from the perspectives of the driver in the car and the bystander the speed of the light coming from the headlight is the same ( 300,000,000 m/s or as physicists say “C”). However with our traditional concept of time, we would expect the bypasser to measure the speed of the light to be:

Because the car would be giving it an extra “speed boost”. Now since everyone agrees on the speed of the light and speed is defined as:

Then both observers (the bystander and driver) must disagree with either:

- The Distance between two points
- The interval of time

Since these are the two values used to measure the speed of the light. These disagreements in time and space intervals are known as time dilation and length contraction respectively (I will address length contraction in another post).

Still confused? think of it this way; the bystander isn’t moving and measures the light to be traveling at speed C. The driver of the car also measures the light and obtains the same value for the speed, How can this be? Well the rulers or clocks that the bystander and driver have must be in disagreement. If there was no disagreement between the clocks or rulers then the bystander and driver would observe different speeds for the speed of the light.

Hopefully This article has had its intended effect in casting doubt on how you perceive time, there will be an additional post with a mathematical proof so that we can quantify this effect.

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