EPT
Einstein's Special Theory of Relativity
Who is this thread for?
The first part is a fairly qualitative description of special relativity and most readers who don't already know this can get something out of it. The second part is a short mathematical analysis of a thought experiment which results in deriving the basic equation of special relativity. It will be tricky for people with insufficient math/physics backgrounds to understand, and won't be that useful for people who already know/understand special relativity fairly well. It will target a somewhat small part of the TL community.
What is meant by Special Relativity and General Relativity?
General Relativity is simply an analysis of the idea that physical laws are the same regardless of the reference frame of the observer (accelerating or inertial). Gravity causes accelerations. The math involved in calculating the influences in different non-inertial reference frames due to gravity is very difficult and I have not studied it... I will not talk about it here. Special relativity is the "special" case where gravity is negligibly weak and all reference frames are inertial (not accelerating). Special relativity is just a subset of General relativity but is much easier to study.
The main postulate of special relativity is that the speed of light is constant in all inertial reference frames. In vacuum, the speed of light is about 3x10^8 m/s. If you were to shoot a photon of light forwards, and run after it at 2.9x10^8 m/s, would the light appear to be getting ahead of you at a rate of 0.1x10^8 m/s?
![[image loading]](http://imgur.com/iRJ9a.png)
You, pictured on top, chasing a photon of light; what would an observer (bottom) see?
The answer is no (by the way you can't actually run that fast). This completely violates the basics of mechanics prior to Einstein. Most students learn in basic physics classes that you calculate relative velocity by adding the velocities of two moving objects in a given direction. You cannot do this with light because most cases would violate the postulate that the speed of light is constant in all reference frames.
So what is actually happening if you shoot a photon of light forwards at 3E8 m/s and then run after it at 2.9E8 m/s? From your perspective (reference frame) the light is traveling away from you at a speed of 3E8 m/s. This might not seem like that big of a deal to you, but consider what another, stationary person (observer) would see if they were watching the event from the sidelines. Wouldn't they see you running forwards at 2.9E8 m/s, and the light traveling forward at 5.9E8 m/s? How can light be traveling so fast from the perspective of a stationary observer?
While you are running you are not "allowed" to see light traveling slower than 3E8 m/s, and the stationary observer is not "allowed" to see light traveling faster than 3E8 m/s. Either we are missing something or you and the stationary observer must exist in different universes (which isn't the case). In order for the universe to make sense there must be something else going on besides what was already identified. The only way this can all make sense, Einstein and others discovered, is if the rate of passage of time is different for you than it is for the stationary observer.
The stationary observer isn't doing anything so let's say the passage of time for him is 'normal.' You are moving (ridiculously and unreasonable) fast, and so apparently the rate of passage of time for you will be different than it is for the stationary observer. Most people know that according to relativity time 'slows down' as you travel faster and faster. Is that consistent with the situation in the picture above?
+ Show Spoiler [answer] +
Yes.
If the rate of passage of time slows down for you, the moving person, then that will give light a 'chance' to get ahead of you. When you are slow everything around you seems fast (many of us have learned that lesson on iccup or battle.net). Since your rate of passage of time is slowing due to your fast movement, light seems like it's moving a lot faster than the 0.1E8 m/s you expected. In fact, it is moving away from you at 3E8 m/s. But the stationary observer's rate of passage of time is normal so light continues to travel from his perspective at 3E8 m/s as he expected. Therefore, in both your reference frame and the reference frame of the observer light is traveling at the correct speed, c, 3E8 m/s.
You might need to reread the past few paragraphs a couple of times to get it if you haven't learned this before; it is quite confusing. I am now going to set up a thought experiment to mathematically calculate the rate at which time slows down as you move with a speed v.
Derivation of Gamma
![[image loading]](http://imgur.com/LyHFc.png)
Common way of demonstrating special relativity
A car is driving to the right with a speed v, as shown in the picture. Mounted to it is a laser pointer (orange) shooting light up towards the ceiling. The ceiling is a mirror and reflects light back down towards the ground. A detector on the car determines how much time it took the light to return to the car after being emitted by the laser. There are two reference frames to discuss:
- You riding in the car analogously to you running in the previous example. Let's call this reference frame #1. Let's call the time it takes light to travel up to the ceiling and back t for this reference frame.
- A stationary observer standing on the ground or somewhere else. Let's call this reference frame #2. Let's call the time it takes light to travel up towards the ceiling and back t' in this reference frame.
Let's make sure we understand what is happening. If you are riding in a car and throw a ball straight up, it should fall straight down and back into your hand, from your perspective. However, from the perspective of a stationary observer on the ground you threw the ball diagonally: up and forwards. Something similar happens with the light. From your point of view (RF#1) the light goes straight up towards the ceiling, and then comes back down to you. The ceiling appears to be moving to the left from your perspective but there is no horizontal component to the light's velocity.
The stationary observer (RF#2) sees the light moving diagonally up and to the right. The horizontal component of the light's speed is v and the vertical component is less than c. The light takes a diagonal path until it strikes the mirror, then reflects. The angle of incidence equals the angle of reflection and the light continues down and to the right with the same pitch.
You might already see a weird inconsistency occurring (just like we did for the first example in this thread). In RF#1 the light travels up a distance D/2 and then down a distance D/2 for a total distance of D. In RF#2 the light travels up a distance D/2, down a distance D/2, but also horizontally to the right the same distance as the car traveled in that time (remember, the light was traveling diagonally in this reference frame). Apparently, from the stationary observer's perspective light traveled further in one trip than it did from your perspective. This means light must have been traveling faster in RF#2 than it did in RF#1. However, according to the main postulate of Special relativity this absolutely cannot be the case (think of our first example). The only way the light could have traveled further in RF#2 despite having the same speed in both reference frames is if time passed more slowly in one reference frame than the other.
The Math
In RF#1 the time it took light to complete one trip, t, is pretty easy to calculate based on the relationship t=d/v. t=D/c. Remember this as we will use it later.
In RF#2 we need to use the Pythagorean Theorem in order to analyze the path of the light. Even though the light traveled in two paths, one before striking the mirror and another after reflecting off the mirror, let's simplify this a bit by saying the light traveled upward a distance D and rightward a distance v * t'. According to the Pythagorean Theorem:
(c * t')^2 = D^2 + (v * t')^2
The first term is the hypotenuse, the second is the vertical component of the right triangle, and the third term is the horizontal component. You need to understand how this equation was put together or you cannot continue. It is not obvious how to use the equation labeled in red above and this one to find a useful relation between t and t' so I will step you through. First, divide both sides of the equation by c^2:
t'^2 = (D/c)^2 + (v/c)^2 * t'^2
Place the rightmost term on to the left side (this is grouping) and then factor out the t'^2 to get:
t'^2 (1 - [v/c]^2) = (D/c)^2
See the red note above; t=D/c so t^2=(D/c)^2. Substitute this into the above equation to get:
t'^2 (1 - [v/c]^2) = t^2
Finally solve for t' and get:
t' = t / sqrt(1 - v^2/c^2) This is the Main Result
This is commonly written as:
t' = t * Gamma
And Gamma is commonly written as:
Gamma = 1 / sqrt (1 - Beta^2)
Where Beta can be written as:
Beta = v/c
This may seem overwhelming but if you follow along and work this all out on paper it should become fairly obvious.
Analysis of the Result
You can determine how much time will dilate (slow down) for the person in the moving car by using this equation. For example, if the car was moving at half the speed of light (ridiculous in practice but good for a mathematical example) then you would find:
t' = t * Gamma = t / sqrt (1 - (.5c)^2/c^2) = 1.1547 t
The difference in the rate of passage of time between the moving person and stationary observer is substantial (more than 15%). At everyday speeds like 30 meters per second or 60 miles per hour relativistic effects are negligible. If you don't believe me calculate t' for the case where v = 0.0000001c.
If I have anything mixed/backwards in my explanations or reference frames then please let me know. If this all makes sense to you and you want to know more about it, then your next step would be to see how time dilation can cause a few interesting things:
- Length Contraction
- Simultaneity (things can actually happen in a different order depending on reference frame)
- Relative Velocities according to special relativity (this came up in Card's Ender series)
The list could go on and on but I did not intend to make an exhaustive guide. I hope many of you enjoyed reading this.
. 
. I wish I took physics instead of engineering
