Shu's Log No. 174: Welcome to Minkowski Spacetime

1905 is known as Einstein's 'miracle year.' In this year, Einstein published three important papers to the world.

These were three papers concerning the 'quantum theory of light,' 'proof of Brownian motion,' and 'special relativity.'

Through special relativity, he discovered that energy and mass are equivalent. That is the famous equation E = MC².

The important point this equation signifies is the conclusion that even if you accelerate, if the mass is heavy, the energy increases, and ultimately the speed of light cannot be exceeded.

The mathematics professor at the Zurich Polytechnic where Einstein studied was Hermann Minkowski. Minkowski described Einstein as a 'lazy dog who never bothered about mathematics at all.'

Upon learning of Einstein's special relativity, Minkowski was astonished and began working on a new research theme. It was to completely describe special relativity through geometry by introducing the concept of four-dimensional spacetime.

According to Minkowski's terminology, a point in spacetime was called an 'event,' and the distance between two points was called the 'spacetime interval.' The one thing that two observers could agree on was the interval or distance in four-dimensional spacetime.

( x , y , z , t )

By specifying this, one can understand 'when,' 'where,' 'what,' and 'what' is happening.

The formula for the distance (s) between two points in Minkowski's four-dimensional space is

S² = (Δx)² + (Δy)² + (Δz)² - (CΔt)²

C represents the speed of light, and Ct represents 'distance.' Δ represents an increment.

Furthermore, when 'normalized' so that the speed C is equal to 1, the formula is simplified to the following:

S² = (Δx)² + (Δy)² + (Δz)² - (Δt)²

Here, if we assume a coordinate graph where the horizontal axis is the variable X representing the three spatial directions (length, width, height) and the vertical axis is 'time',

S = √|X² - t²|

In the case of a beam of light, the distance (X) is 'Ct', but since we set C=1,

X = t

In other words, when X = t, the distance light travels through spacetime '√|X² - t²|' is always 0. (*One just has to get used to this trick)

This shows that distance in Minkowski spacetime is different from distance in Euclidean space.

In the following graph (figure), the sum of the lengths of line segment AD and line segment CD, that is, <2√|X²-t²|>, is shorter than the straight-line distance of line segment AC.

Minkowski also introduced the <spacetime diagram> as a way to geometrically visualize the structure of spacetime.

When X=t, the 45-degree line passing through point (1, 1) and point (-1, -1) becomes the path that light always travels in Minkowski spacetime.

And when you add a second spatial dimension (the x-axis and y-axis, with the t-axis orthogonal to them) and rotate that V-shape around the origin, it becomes a surface called a <light cone>. That is this figure.

This figure would later be used extensively in Penrose and Hawking's research on black hole <singularities>. As an example, I will provide a figure of a <model of a black hole created by matter collapsing at the speed of light>.

Incidentally, his student Einstein disliked Minkowski, calling it 'superfluous learning'.

However, in order to arrive at the general theory of relativity for gravity, Einstein eventually had to apply geometry, so he sought help from his friend and mathematician at ETH Zurich, Marcel Grossmann.

The words he uttered at that time were, 'Otherwise, I will go mad.'

The approximately 10 years from special relativity to general relativity were a struggle for Einstein with unfamiliar mathematics.

In particular, understanding the curvature tensor (a framework for dealing with curvature) of Riemannian geometry, which deals with space, became the final obstacle.

Many mathematicians were involved in the completion of general relativity. This essay has drawn heavily from books by mathematicians who focused on the mathematics that supports physics, and sometimes the mathematics that has led physics.

For black holes, I recommend books written by Hisaaki Shinkai. (Figure)