Space-time diagram

As shown in the “What is spacetime ?” article, we live in 4 dimensional spacetime. Doesn’t matter whether you are at rest in your frame, you are moving along the time axis. But one more thing to note, if you are at rest in your frame, it does not mean that you are at absolute rest. You may be in motion in some other frames. And similarly those frames are in motion in your frame. These are the major concepts of “Special theory of relativity” and should be understood and visualized properly. And the best way to describe motion is to draw spacetime diagrams. It is the easiest way for a beginner to understand the results of “Special theory of relativity”. But how to draw spacetime diagrams? This is actually very easy. As it is impossible to draw all the 4 axes (time axis, x axis,y axis,z axis) here we will consider only the time axis and the x axis. The below diagram shows a two dimensional slice of spacetime, the t-x plane.

Explanation: Suppose this is your frame and at time t = 0 you are at the position 𝓞 and a cat is at 𝓐. As shown in the diagram, the cat moves as time passes. At a particular time t = t, the cat is at some point 𝓟. Clearly the position of the cat is a function of time. If the cat is moving with uniform velocity, say at velocity v with respect to you, the position of the cat is x = vt. This is an equation of a straight line and the line is shown in the diagram( line AP). The diagram thus describes the motion of the cat in your frame. This is called the “world line” of the cat. Similarly everything at rest or in motion has its world line in your frame. The time axis is your world line as you are at rest in your frame and you are moving along the time axis. So, x is always zero for you. If another cat (which is at the point 𝓑 at time t = 0) is at rest in your frame then its world line is parallel to your time axis and passes through the point 𝓑. Every point in this diagram is called an “Event”. So the points 𝓑, 𝓐, 𝓟, 𝓠, all are events which have a constant time and position as coordinates. Note that by saying constant position I mean the position of the point at that corresponding time, it doesn’t mean that something at that point is at rest. Events are actually the points having some positions at that corresponding moment.

Hope you have understood the basics of drawing a spacetime diagram. Now let’s move to the rules and convention of drawing a spacetime diagram. Every point or “Event” has four coordinates, the first coordinate is time coordinate, second, the x coordinate, third, the y coordinate and the last coordinate is the z coordinate. For example, say the position of the first cat at time t = 0 is (0,5,0,0). One more thing to note, here time and all the other special coordinates are measured in meter as described in the “can the speed of light be exactly 1?” article. So if you have not read the article first read it because all the further calculations are based on this system. The world lines of particles in your frame depend on their velocity relative to you. So the world line of the cat is the line x = vt in your frame. This implies that the slope of the line is 1/v. Similarly the slopes of all the world lines are the inverses of their velocity in your frame. So, the slope of the world line of light is 1/c i.e. 1(as in this measurement system c is exactly 1). As nothing can move faster than light, so no world line can have slope less than the slope of the world line of light.

Conventions:

• Events are denoted by cursive capital letters, however the letter 𝓞 is reserved for observers.
• The coordinates are called (t,x,y,z)
• The coordinates can also be given alternative names such as (x⁰,x¹,x²,x³). The superscripts are not exponents, these are just labels, called indices. So x⁰ denotes the time coordinate, x¹ denotes the x coordinate and so on. (x²)² denotes the square of the y coordinate. Generically, the coordinates x⁰,x¹,x²,x³ are referred to as x^α where 𝛼 can take values from {0,1,2,3}. A Greek index is used to denote the coordinates. If 𝛼 is not given a value, then x^α is any of the four coordinates.
• We may want to distinguish between t on the one hand and (x,y,z) on the other hand. We use Latin indices to refer to the spatial coordinates alone. So a Latin index (e.g. i,h,j,k,l) will be assumed to take a value from {1,2,3}. If it is not given any value then xⁱ is any of the three spatial coordinates.