Relativity, the math, yeah, really, learn it, cry over it, love it, not necessarily in that order

We are bounded in a nutshell of Infinite space: Blog Post #33, Free Form #5: Relativity, the math, yeah, really, learn it, cry over it, love it, not necessarily in that order.

Following up the earlier blog post in September (http://ay17-rcordova.blogspot.com/2015/09/relativity-yes-we-are-doing-this-it-is.html), relativity is an essential part to how we’ve come to understand how the universe works, especially when considering the fact that the largest objects and forces in the universe move at speeds comparable, if not at, the speed of light. So, we shall begin with the fundamentals of how Special Relativity, as told to you by one who learned it not a month ago.

In Special Relativity, there are two basic assumptions which make up the basis of all the equations:

1.       There are no preferred inertial frames

2.       The speed of light is the same in every inertial frame

From these two basic tenets, there are three Fundamental effects that can be observed, and have been consistently tested.

The First of these fundamental effects is the Loss of Simultaneity. Here, two events that coincide in one frame (that is moving) do not coincide in another frame of reference. Take the example of a moving train, in which two flashlights placed at the center send photons to the opposite ends of the train. Within the train, they hit the ends at the same time, but because of the additional speed of the train, an outside observer would perceive the photons hitting the front of the train before the back end. This effect is paralleled by the next one, and we’ll go into the math with it.

This loss of simultaneity leads us to the Second Fundamental effect, Time Dilation. If you click on the link to see the post from September, the classic example of this effect is described by the observer on the train’s perception of light and the observer on the ground’s perception.

“The classic example Einstein gives in his papers on Special and General Relativity is the case of two persons, one on a train heading in a direction towards a point where a lightning bolt just hit the ground, and another person stands some distance away and can observe both the train and the lightning bolt.  In this scenario, the person on the train is moving at a speed v, who will hit the rays of light with this speed and thus (one would expect to) perceive the speed of photons as the intrinsic speed of light minus the speed of the person. This would differ from the person outside the train, seeing the lightning bolt come and thus (expect) to perceive light at its normal speed.

However, the speed of light must be CONSTANT at all times, so the person on the train must have something change in order for him to experience the speed of light at the correct value. What happens is, as Einstein describes in Special Relativity, that the person moving with speed v experiences time (and length and differently, it slows down in his (moving) reference frame and so the speed of light he perceives is maintained at the constant rate. For the person outside the train, he would see the light coming from the lightning bolt at its normal speed, without any special considerations to be taken into account. But as he sees the light going towards the train in the distance, he would clearly see the light in direction of the train has the same speed as the light that reached his reference frame, maintaining the constant speed of light and set consolidated with the perception of the person on the train, instead of two people experiencing two different speeds of light.”

Therefore, you can assume each person had a clock on them at the time of the lighting strike, and each perfectly measured the time at which they saw the light, and each would have different measurement. You can also picture this by seeing how two different paths light takes take different times to be completed, indicating that the longer path, the one where the frame itself was also moving. This is illustrated by:

                                 

As you can see, the initial case projects a different (and shorter) path once in the reference frame of A, than in the frame of B. So if the original distance was simply 2h, and the speed travelled c, so the time for the photons to traverse is \[\frac{2h}{c}.\] In the case of B, the total vertical distance the photons travel is 2h, but the speed component is different, being (by use of a bit of reasoning with the Pythagorean Theorem): \[\sqrt{c^2 – v^2},\] so the time would be: \[\frac{2h}{ \sqrt{c^2 – v^2}}.\] These two times can be compared and set as: \[t_A \sim t_B,\] \[ \frac{2h}{c} = \frac{2h}{ \sqrt{c^2 – v^2}},\] \[ 1 = \frac{c}{ \sqrt{c^2 – v^2}},\]\[ t_A = t_B \frac{1}{ \sqrt{1 – \frac{v^2}{c^2}}},\] where the additional factor for \(t_B\) is what is called \(\gamma_v\):\[\gamma_v = \frac{1}{\sqrt{1- \frac{v^2}{c^2}}},\] a common “conversion” factor which allows the switching of frames in Special Relativity. So the final time dilation equation is \[t_A = \gamma_v~ t_B, \] or \[t_B = \gamma_v~ t_A, \] depending on the reference from which the action is being observed.

This principle of time dilation, the factor \(\gamma_v\), establishes the base for most Relativistic problems (as long as they don’t involve gravity). 

The Third Fundamental Effect is Length Contraction. This effect establishes how an object moving at a speed is contracted proportionally to the increase in speed. This is described by \[L^\prime = \frac{L}{\gamma_v},\] describing how length is inversely proportional to speed.

These three fundamental effects are at the core of relativistic situations, which are then expanded upon by Einstein in 1915 with his treatise on General Relativity (100 Years of General Relativity has been floating around Science Media the last few months, if you hadn’t heard), where the impact of gravity is coalesced with relativity to describe myriad situations like Gravitational Lensing and Microlensing (see the posts about that by going back to the September group). These equations have become the basis for modern physics and astrophysics, forming part of the ongoing search to unite quantum mechanics and General Relativity, the greatest descriptions of the universe humans have ever had the ability to (sort of) prove.