The next few posts in this series will introduce concepts like Relativity, Black Holes, and Hawking Radiation. These are sections from a paper I have been working on that will introduce such concepts to high school students not just in a pop-science way, but in a more substantial and mathematical way.

If a student is interested in these topics and tries to read about them, they are encountered with two options. Either they read an over-simplified version written in a pop-science article, or, they try to read the actual papers where they see many complex equations and difficult words which discourages them from further reading.

Therefore, they are left with a passion to understand but no means. There are many subjects that need to be understood before tackling high-level questions like Special and General relativity. Understanding these subjects takes a lot of time and more importantly not all topics in these are important for understanding what we have sought out to understand. The primary purpose of this paper is to pick and choose important topics from these subjects so that one is able to understand the complicated concepts without any prior experience in the field and the assumed level of expertise of an enterprising high schooler.

An understanding of the following topics are needed to understand the high-level concepts of Hawking Radiation and Black Hole Physics:

  1. Special Relativity (1) – Postulates, Diagrammatic representations and Invariance principles
  2. Special Relativity (2) – Vector Algebra, Four Velocity and Momentum:
  3. General Relativity – Killing Vectors, Curvature and Geodesics
  4. Black Holes
  5. Hawking Radiation

The next few posts in this series will cover these concepts and have been taken from my paper.

This post covers the first chapter: Special Relativity (1) – Postulates, Diagrammatic representations and Invariance principles.

Special Relativity (1) – Postulates, Diagrammatic Representations and Invariance Principles:

(These derivations and definitions have been made using the help of [1].)

Postulates of Special Relativity:

  1. Principle of relativity: No experiment can measure the absolute velocity of an observer. the results of any experiment performed by an observer do not depend on its speed relative to observers not involved in the experiment.
  2. The universality of Speed of Light: The speed of light is always a constant with respect to any unaccelerated observer in any frame. (3*108 m/s)

Spacetime Diagrams:

First, let me introduce the concept of Spacetime Diagrams. These give a graphical representation of events and observers. This powerful tool will be very useful to us in our studies.

Suppose we have a 2d coordinate axis with the x-axis representing the x coordinate and the y-axis representing the time coordinate, as given below: 

A singular point in this diagram is called an event. A line represents position as a function of time or vice versa. It is called the “world line” of the particle.

We also use different units of time in these diagrams. For the sake of convenience, we will use such a system of units so that the speed of light becomes 1. We define something called “1 meter of time”. It is defined as the time taken by light to travel 1 meter of distance.

There are a few rules that have to be followed while making spacetime diagrams:

  1. Events are denoted by capital letters.
  2. Coordinates are of the form (t,x,y,z)
  3. Alternative coordinates are generally used for convenience. This means that (t,x,y,z) is represented as (x0, x1, x2, x3). 0,1,2,3 are not exponents, they are just indices used to express the coordinates.
  4. Latin indices are used for the spatial coordinates. So xi can represent any
  5.  spatial coordinate.

Let us now take the example of time dilation to have some more practice in spacetime diagrams:

As seen in the diagram below, it is found that when the clock moves in the axis it reaches event A in 1 unit time but in the t axis, event B has coordinate t = 1/(1-v2). (which runs slower)

(More detailed explanation in chapter 1 of [1])

This is a good example of how space-time diagrams are used to derive important results.

Spacetime diagrams will be used extensively in black hole physics also. They are used a lot for drawing the horizons of a black hole and finding its event horizon.

Making the Coordinate system of another observer:

An observer can be defined using a coordinate system in spacetime. We can draw the coordinate system of one observer with respect to another observer.

Let’s say we have an observer θ’ moving with velocity v in the positive x-direction of another observer θ. t’ is the time axis of θ’. Its spacetime diagram is given below:

We shall now see how these axes behave in a practical example:

Look at the diagram below:

In this case, we are looking at a light ray travelling from -a on the t’ axis to the x’ axis and being reflected onto +a on the t’ axis again, from the point of view of θ’.

If we look at it with respect to θ the following is seen:

Since we are using the new units of velocity, light will have a velocity of 1 and hence will move in a 45* line.

We see that at event E it is released from a point -a on the t’ axis. It reaches the x’ axis at event P and must be reflected along the same 45* angle in the other direction and reaches the t’ axis again at R. The point of reflection P is shown in the diagram. The line joining P to the origin is the x’ axis and is seen to not coincide with the x-axis.

Due to the second postulate of special relativity, the speed of light is always 1 and always makes an angle of 45*.

This shows that events simultaneous to θ’ are not simultaneous to θ.

This is a very important feature of Special Relativity.

Invariance of Time Interval:

Let us consider two events E and P from the previous figure. In frame θ, since the speed of light is 1 the differences in coordinates of E and P satisfy the relation:

But due to the universality of the speed of light, this is true for frame θ’ also. So:

For some events that may be on or not on a world line of a light beam, the interval of two events is defined as:

If the interval square in frame θ (Δs2) = 0, it implies that (Δs’ 2) is also equal to 0.

From this we can derive that:

Δs2 = Δs’ 2

And hence, we are able to prove that the interval between two events is not dependent on which observer is calculating it:

Δs2 = Δs’ 2

(A detailed derivation is found in Chapter 1, Section 1.6 of [1])

Therefore, we have proved the invariance of an interval.

There is another concept of light cones.

If Δs2 is positive (this means that the spatial elements dominate) it is called spacelike separation. If it is equal to 0 it is called null separated and if it is negative it is timelike separated.

All the points that are inside the cone are timelike separated and can be reached with velocities less than the speed of light. The points on the cone are null separated and move at the speed of light. The ones outside the cone are similarly spacelike.

Lorentz Transformations:

The Lorentz transformations give a way to get the coordinates of events in the θ’ frame when the coordinates of an event in frame θ are given.

These are very important and are used to convert coordinates from one frame to another.

These are the Lorentz Transformations:

A detailed derivation of these can be found in Appendix 1 of [2].

References:

[1]: Bernard Schutz, ‘A First Course in General Relativity’, Cambridge University Press, 1985

[2]: Albert Einstein, ‘Relativity: The special and general theory’, Fingerprint Classics, 1916

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