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Lightcone Structure of Spacetime

I want to start talking about some of the more philosophical aspects about physics but it won't be possible without going over important stuff. For this post I want to explain the lightcones that physicists use in relativistic physics. If you're already familiar with the concept of natural units you can go ahead and skip to the part labeled "lightcone structure of spacetime" and my feelings won't be hurt. But it may also help to watch Sabine Hossenfelder's video first. These are highly important to pin down in order to understand any branch of theoretical physics and what I discuss here is no exception. 

Lightcone Structure of Spacetime

In mathematics and physics the metric of spacetime allows us to calculate the shortest distance between two points, the famous metric equation for Euclidean geometry and space is is the Pythagorean theorem for a flat two dimensional triangle

The theorem can be extended in three dimensions by adding a third dimension of space for a 3-dimensional universe but in special relativity this is still not enough. Space and time are combined into a single entity called "spacetime" so we need to modify the metric to include a variable for time as we have one dimension of time and three dimensions of space. We can easily do this but time is distinct from space. So our operation should be a negative rather than a positive sign

For a geodesic in a spacetime described by this metric the proper time between two points is taken to be the integral 

In special relativity the metric only describes flat spacetimes and doesn't add the scale factor or take into account any gravitational effects. For particles moving much slower than the speed of light their motion is well approximated by the physics of Galileo and Newton. But according to general relativity, however special relativity is wrong globally but correct in local regions of spacetime. In infinitesimal regions our spacetime is described as "Minkowskian" and described by the metric given above. This metric allows for three possible solutions 

In this case we interpret the two events as being separated through a distance just like ordinary Euclidean geometry and we say they are "space like" separated.

In this case our solution is less than zero and what's under the square root is an imaginary number. So we flip the sign and interpret the answer as a time value and we say these events are "time like" separated from one another.  

In this case the events are not separated through time or space and we say they are "light like" separated. These different possible values are connected as different regions of a lightcone, the future and the past lightcone are regions which are time like separated from us whereas the region outside of the lightcone are regions which are space like separated from us. Only events which you can send or receive a light ray from are included in your lightcone. For practical purposes we ignore the present.

Any three dimensional space like surfaces in which the elements of the collection is labeled by real numbers and increases steadily from one surface to the next can be used to define what is "present" what is "past" and what is "future" relative to your reference frame. Each event on one surface is simultaneous to the events on that same surface. This kind of structure might lead one to believe that time is a block rather than just the present but that's a rather controversial topic for another time.


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