Pythagoras's Theorem In 4D - The Space-time Interval
The Space-time Interval
We have found that from assuming the speed of light is an invariant; that time, distance and the order of events is not invariant. We now want to see if we can find some things that do not change no matter what frame of reference you are in.
An example of this in classical mechanics is distance. We can all agree on the distance between two objects. No matter who's coordinate system we use; the distance between two objects will always be the same and this can be calculated in 2D space using Pythagoras's theorem. Distance = √(x2 + y2) Any rotation or translation of the x and y axis will give this same distance.
In special relativity there is a quantity similar to distance in classical mechanics called the space-time interval/distance. This is invariant under the Lorentz transformations. Like in previous posts, we will work with 1 spacial dimension.
𐤃s2 is the space-time interval
𐤃s2 = (c𐤃t)2 - (𐤃x)2 is the space-time interval for a stationary observer
𐤃s'2 = (c𐤃t')2 - (𐤃x')2 is the space-time interval for a moving observer
𐤃s2 = 𐤃s'2 - The space-time interval is the same no matter what perspective you look from
(c𐤃t)2 - (𐤃x)2 = (c𐤃t')2 - (𐤃x')2
What this means is if we have a stationary observer and they calculate the difference in time and difference in space between 2 events. They can calculate the space-time interval.
Any inertial observer will calculate this same space-time interval despite time dilation and the relativity of simultaneity affecting the difference in time and length contraction affecting the difference in space. The squared terms in the equation may remind you of Pythagoras's theorem which gives distance in 2D or 3D space (also an invariant).
So what does this mean physically? Right know this is just a bunch of equations? We will explore what this actually represents in the next post on special relativity. It will become clear after we explore how causality works in special relativity.
Proof that the space-time interval is invariant under the Lorentz transformations i.e. it is the same for all inertial observers
Thanks for reading. If you enjoyed this post or any of my others, follow and subscribe to my blog. Feel free to discuss anything related to this post or ask questions in the comments below.
Check out my other posts on special relativity! (Link to all the posts)
How To Travel To The Future
How Galilean Relativity And The Lorenz Maxwell Equations (The 2 Postulates) Contradict
The Postulates Of Special Relativity
a1*a1 + a2*a2+ a3*a3 .... + an-1*an-1= an*an true ???? n almost infinity
ReplyDelete