What Makes Special Relativity So “Special”?

Hello, and welcome back to MPC! Last week, we analyzed the relationship between spacetime diagrams, light cones, and causality. This week, we will take a step back and think about special relativity as a whole.

What is special relativity? Recall that we initially defined special relativity with two principles:

  • The laws of physics are the same in all inertial frames of reference.
  • The speed of light is constant in all inertial frames of reference.

Beyond those principles, though, let’s just think about its name: “special relativity.” Why is it called that? Well, we know what relativity is from a previous post, and it is rather clear how special relativity incorporates our definition of relativity. For example, when we spoke about time dilation, we discussed how one person’s clock would appear slower to you if he were moving relative to you. Moreover, the exact sluggishness of that person’s clock would be dependent on how fast he was moving relative to you. Overall, this idea of relative motion played a huge role in special relativity and our definition of space and time.

Alright, so the “relativity” part of special relativity makes sense, but what about the “special” part? Yes, special relativity is exciting and mind-blowing, but special? Electricity is pretty exciting too, but we don’t call it “special electricity.”

The truth is, special relativity is special because it can only model space and time in a very specific, or special, scenario. Recall the scenario we used to derive the time dilation formula (in this post): we had a person inside of a rocket ship that was zooming off at a speed of v:

1

Figure 1: The scenario we used to derive the time dilation formula

From this simple scenario, we were able to derive a powerful formula:

Eq1.PNG

However, in deriving this formula, we made a rather large assumption: the rocket ship was moving at a constant speed. How so? We explicitly stated that the rocket ship was traveling at a speed of v! This is okay, but it does present a “small” problem: our derivation does not account for the scenario where the rocket ship’s speed is continuously increasing throughout its journey. Recall that acceleration is the change in velocity per unit time (see this post). Acceleration is not a foreign concept to us — we are used to acceleration when driving. When you are stopped at a traffic light and the signal turns green, you step on the gas pedal to accelerate your car to the speed limit. Let’s imagine our rocket ship does something similar: it starts at a speed of 0, and accelerates to a higher speed, say 0.98c. How slow does the person inside of the rocket ship’s clock seem to the person on the ground? Well, let’s just calculate the relativistic factor (see this post):

Eq2.PNG

But wait, what is v? The rocket ship’s speed is constantly changing: at one point it isn’t moving, later it is traveling at 0.1c, later it is traveling at 0.2c, etc. Which speed do we plug in?

This is why special relativity is so special: it only works for objects moving at constant velocity (not accelerating). Special relativity cannot handle acceleration!

So what do we do if we have objects that are accelerating?

Who cares about objects that are accelerating? Is acceleration even that common in nature?

Yes, it is! Acceleration is everywhere! Drop a book and it accelerates to the ground, roll a ball on the floor and it decelerates to a stop! If we want to understand our universe, we need some way of analyzing space and time when acceleration is involved.

Einstein, the father of special relativity, realized this limitation of his theory and worked hard to make special relativity more “general” (account for all types of motion, with or without acceleration). The result of his efforts, general relativity, took many years to develop and is much more complicated than special relativity (as we will soon see; do not worry though, it is nothing we cannot handle!).

Next week, we will be diving into general relativity. We will be sticking to conceptual discussions rather than mathematical discussions (because of general relativity‘s mathematical complexity). I hope you are excited — I certainly am! See you then!

(featured image: https://www.walldevil.com/wallpapers/a55/background-computer-apples-desktop-theme-wallpapers-food.jpg)

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