Thus was born Einstein's equation, a mathematical formula that describes the universe! What does “bending time” (Hideki Asada) mean?

Detecting mysterious gravitational waves as distortions in outer space. A popular book presenting the latest in cosmology that approaches the birth of the universe from this observational fact that shocked the world.How did the universe begin with nanohertz gravitational waves and the physics of the birth of the universe?“. In this article we will discuss “gravity”, which is considered one of the biggest mysteries in modern physics, and look at how to create “Einstein's equations”, a mathematical formula that Einstein devised to describe the universe. I will go.

*This article is based on “How did the universe begin?(Bluebacks) has been rebuilt and re-edited.

What is the concept of Einstein's equations?

One anecdote related to gravity is the experiment of falling from the Leaning Tower of Pisa, said to have been performed by Galileo Galilei. This is the fact that objects fall with the same acceleration regardless of their characteristics.

As introduced in a previous article, the degree of curvature of spacetime is the geometry of spacetime, so it is consistent with Galileo's experimental results of falling with the same acceleration.

So, what determines the degree of curvature of space-time?

There is a well-known mathematical formula that shows that the energy and momentum of matter are conserved.

Einstein noted that this formula and an identity in Riemannian geometry (a relational expression that always applies between variables) called the “Bianchi identity” have the same form.

From here he arrived at the “gravitational field equation,” which was later known as Einstein’s equation.

The concept of Einstein's equation

This is where the new word “gravitational field” comes into play. When we say gravity, we are referring to a “force” that acts on something. Since this force affects the body at every point in space, the space in which this force acts is called the “gravitational field.”

In fact, at the same time, Davit Hilbert, a great mathematician at the University of Göttingen (Germany), was also looking for an equation for the gravitational field, but Einstein won the competition.

Einstein's equations are equations that relate the distribution of matter to the geometry of spacetime (which means gravity). Assuming the distribution of matter, we can create an equation for the gravitational field (the geometry of space and time), and by solving that equation, we can find the gravitational field.

To describe the curvature of space-time

Now, what is important here is that the degree of curvature cannot be measured in one place.

For example, a point on the Earth's surface appears just flat. This fact is closely related to the “equivalence principle” in general relativity. The equivalence principle here means that even if spacetime were curved, at any point in that spacetime, physics where there is no curvature, namely the special theory of relativity (theory of relativity excluding gravity), means it would work. Therefore, to discuss the degree of curvature, it is necessary to compare that point with the surrounding points.

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The geometry obtained by formulating this comparison work mathematically is “Riemannian geometry.”