Physics for everyone

Geodesics - from intuition to equations

The simplest kind of geometry, taught in schools, is the so called Euclidean geometry - named after an ancient Greek mathematician, Euclid, who described its basics in the 4th century BC in his "Elements". It is based on the notions of points, straight lines and planes and it seems to correspond perfectly to our everyday experiences with various shapes. However, we can notice problems for which Euclidean geometry is insufficient even in our immediate surroundings.

Let's imagine, for example, that we are airline pilots and our task is to fly as quickly as possible from Warsaw, Poland to San Francisco. We take a world map and knowing from Euclidean geometry that a straight line is the shortest path between two points, we draw such a line from Warsaw to San Francisco. We're getting ready to depart and fly along the course we plotted... but fortunately, our navigator friend tells us that we fell into a trap.

The trap is that the surface of the Earth isn't flat! The map we used to plot our straight line course is just a projection of a surface that is close to spherical in reality. Because of that, the red line on the map below is not the shortest path - the purple line is:

Red line - a straight line between Warsaw and San Francisco on the map. The purple line is the actual shortest path.

Lorentz transformations, light cones

In the previous article:

  • What are events and spacetime?
  • What are world lines?
  • Simple spacetime diagrams
  • How does the inseparability of space and time influence their perception by observers?

Most of the illustrations in the last article used rotations, but it turned out eventually that rotations aren't the correct transformations that would let us look at the spacetime from the point of view of different observers. Now we will take a look at transformations that actually describe reality - the Lorentz transformations.


Events and space-time

The first entry in the series will be quite basic, but I think that some problems will nevertheless be quite interesting. We'll be talking about what is the space-time, events, and we will show where the theory of relativity comes from. So, let's go :)

The notion of space-time is briefly mentioned at school, but usually the profound consequences of combining space and time into a single entity aren't explained too much. To understand this, one must first go a bit deeper into the details of this idea.