Special Relativity

In 1905, Einstein was a 26-year-old university graduate who was working in a patent office. He enjoyed producing groundbreaking physics work in his spare time, and in 1905, he produced his theory of special relativity. This theory has two parts. First, it says that the laws of physics are identical throughout the universe for any “inertial” (non-accelerating) observer. Second, it says that the speed of light is the same for all observers. These may seem like commonsense, innocuous statements, but they have profound implications. As you will see, while some things are absolute, such as the speed of light, other things are relative based on the observer, such as space and time.

Own work, created with Microsoft Powerpoint

Imagine that you are in a vintage race car traveling 70 miles per hour to the east, and your buddy’s jalopy is traveling at 30 mph to the west. In this case, your buddy’s car is traveling towards you at 100 mph. This is because YOUR car is not traveling away from you at all – it is traveling at a lowly 0 mph relative to you. Likewise, from your buddy’s perspective, his car is not moving at all, but your car is moving towards him at 100 mph. At first, this seems like a paradox: how can something be moving at 0 mph and 100 mph at the same time? Of course, it all makes sense when you take who’s driving the car into account. This illustrates the concept of inertial reference frame, where any arbitrary, non-accelerating object can be defined as being stationary while other objects are whizzing all around in every direction. I may think I’m stationary sitting here writing this tutorial, and I am from my reference frame, but try asking the man on the moon if I’m stationary. Heck, he can only see me for half of the day!

Weird things happen when you travel at speeds close to the speed of light. Say, for example, that I am flying in a spaceship at the speed of light to the east, and my buddy is flying at the speed of light to the west. It would seem, then, that from my inertial reference frame, my buddy is traveling at two times the speed of light away from me. In the graphic below, ‘c’ stands for the speed of light, while ‘2c’ stands for twice the speed of light.

Own work, created with PowerPoint

Own work, created with PowerPoint

However, this is not possible. Due to the theory of special relativity, since we are both non-accelerating objects, the same laws of physics apply for us, and the speed of light is the same for us. Nothing can travel faster than the speed of light. Therefore, my buddy is only traveling toward me at the speed of light, not twice the speed of light.

In the graph below, the red, dashed line represents the relative speed of one rocket from another (in terms of its velocity divided by the speed of light) if we just take the sum of the parts like we did with the cars. The blue line shows the relative speed if we take special relativity into account, like we did with the rockets. There isn’t much of a difference until after v/c=.1, meaning that the velocity (v) is 1/10th the speed of light. One tenth the speed of light is 66,960,000 miles per hour! So yes, relativity affects us all, but the effects are so small that it is impossible for us to notice them. Still, it’s fascinating to know that any movement has an effect on time and space from your reference frame.

Credit: Stack Exchange

Credit: Stack Exchange

Time Dilation and Length Contraction

In the car example, the speed of my car was completely dependent on reference frame. I could’ve been going 0 mph or I could have been going 100 mph. If there was a hitchhiker standing on the side of the roadway, he would have said I was going 70 mph. And as it turns out, time, just like speed, position, and so many other things, is relative.

If I’m traveling at light speed towards from my buddy and he doesn’t appear to be going anywhere, then time is going normally for me, but no time has passed for him. If time was going for him, he would be moving towards me at a speed greater than the speed of light, which is not possible. The lengthening of time for the object/person the observer sees in their inertial reference frame is a phenomenon known as time dilation.

A consequence of time dilation is length contraction. If time passes more slowly for other objects flying by the observer than the observer himself in his inertial reference frame, then the length of those objects the observer sees must contract in order to satisfy the relationship between time, length, and the speed of light. For example, let’s say that my buddy and I choose to measure lengths in how long it takes light to cover a certain distance (ex: lightyears). We can express the length of a ship as lightyears=(speed of light)*time, so in the above example where the time passed for my buddy approaches 0 from my inertial reference frame, the length of his ship must also approach zero from my inertial reference frame. For more on time dilation and length contraction, click here.

All of this stuff means is that space and time are not separate entities. They are intrinsically woven together, and we call this entity space-time. Thankfully, these effects only become noticeable when traveling at “relativistic” speeds… i.e., speeds approaching the speed of light. Otherwise, I imagine everyday communication would be quite difficult!

Light Cones

2-Dimensional Light Cone with a Time Dimension

2-Dimensional Light Cone with a Time Dimension
SVG created by Wikimedia user K. Aainsqatsi, original PNG created by Wikimedia user Stib

The fact that nothing can travel faster than the speed of light has important consequences for an observer, and these can be illustrated via “light cones.” A light cone diagram plots space in two dimensions and time in one dimension and has past and future light cones originating from the location of the observer. As you go further forward or backward in time, the radius of the light cone expands, where the radius is equal to the speed of light multiplied by the number of seconds away from the present the observer is.

Since nothing can travel faster than the speed of light, anything outside of the light cone cannot affect or be affected (or even observed) by the observer. In the past light cone, the observer can observe certain events in the past, such as a gamma ray burst from a galaxy 13 billion light years away or a solar flare that occurred on the sun 8 minutes and 20 seconds ago (the time it takes for light from the sun to reach the Earth, assuming Earth is 93 million miles from the sun). In the future light cone, the observer can be observed by other observers in the future.

There are other consequences of special relativity, a major one being that events that may appear simultaneous to one observer may not appear simultaneous to another observer. Also, weird stuff happens when you are accelerating (i.e. not in an inertial reference frame). It’s all incredibly confusing and counter-intuitive.

That’s it for special relativity! Click on the hungry black hole below if you want to learn about Einstein’s magnum opus, his theory of general relativity.

A black hole orbiting a star as part of a binary system and destroying it in the process. Credit: NASA

Written by Charlie Phillips – charlie.weathertogether.net. Last updated 12/1/2017