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How to understand Einstein's equation for general relativity - Big Think
Sep 15, 2021 7 mins, 26 secs

Countless scientific tests of Einstein's general theory of relativity have been performed, subjecting the idea to some of the most stringent constraints ever obtained by humanity.

Although Einstein is a legendary figure in science for a large number of reasons — E = mc², the photoelectric effect, and the notion that the speed of light is a constant for everyone — his most enduring discovery is also the least understood: his theory of gravitation, general relativity.

But Einstein's conception was entirely different, based on the idea that space and time were unified into a fabric, spacetime, and that the curvature of spacetime told not only matter but also energy how to move within it.

This fundamental idea — that matter and energy tells spacetime how to curve, and that curved spacetime, in turn, tells matter and energy how to move — represented a revolutionary new view of the universe.

Put forth in 1915 by Einstein and validated four years later during a total solar eclipse — when the bending of starlight coming from light sources behind the sun agreed with Einstein's predictions and not Newton's — general relativity has passed every observational and experimental test we have ever concocted.

Yet despite its success over more than 100 years, almost no one understands what the one equation that governs general relativity is actually about.

Einstein's original equation relates spacetime curvature to the stress-energy of a system (top).

These five terms, all related to one another through what we call the Einstein field equations, are enough to relate the geometry of spacetime to all the matter and energy within it: the hallmark of general relativity.

A mural of the Einstein field equations, with an illustration of light bending around the eclipsed sun, the observations that first validated general relativity back in 1919.

The Einstein tensor is shown decomposed, at left, into the Ricci tensor and Ricci scalar.

You might be wondering what is with all those subscripts — those weird "μν" combinations of Greek letters you see at the bottom of the Einstein tensor, the metric, and the stress-energy tensor.

E = mc² is a scalar equation because energy (E), mass (m), and the speed of light (c) all have only single, unique values.

In general relativity, the fact that we have four dimensions (three space and one time) as well as two subscripts, which physicists know as indices, means that there is not one equation, nor even three or four.

In general relativity, space and time are continuous, with all forms of energy contributing to spacetime's curvature.

The "Ricci" part is volume distorting, and that plays a role in the Einstein tensor, as the Einstein tensor is made up of the Ricci tensor and the Ricci scalar, with some constants and the metric thrown in.

The "Weyl" part is shape distorting, and, counterintuitively enough, plays no role in the Einstein field equations.

The Einstein field equations are not just one equation, then, but rather a suite of 16 different equations: one for each of the "4 × 4" combinations.

This framework, in many ways, takes the concept of a differential equation to the next level.

A differential equation is any equation where you can do the following:.

The Einstein field equations allow you to relate spacetime curvature to matter and energy, in principle, for any distribution you choose.Credit: JohnsonMartin / Pixabay.

Only, when we begin dealing with general relativity, it is not just one equation or even a series of independent equations that all propagate and evolve in their own dimension.

Instead, because what happens in one direction or dimension affects all the others, we have 16 coupled, interdependent equations, and as objects move and accelerate through spacetime, the stress-energy changes and so does the spatial curvature.

First off, the Einstein tensor is symmetric, which means that there is a relationship between every component that couples one direction to another.

The power of this part allows us the freedom to choose whatever coordinate system we like, which is literally the power of relativity: every observer, regardless of their position or motion, sees the same laws of physics, such as the same rules for general relativity.

In general relativity, those conserved quantities translate into energy (for the time dimension), as well as momentum in the x, y, and z directions (for the spatial dimensions).

Even though it is impossible to define things like "global energy" overall in general relativity, for any local system within general relativity, both energy and momentum remain conserved at all times; it is a requirement of the theory.

Another property of general relativity that is different from most other physical theories is that general relativity, as a theory, is nonlinear.

If you have a solution to your theory, such as "what spacetime is like when I put a single, point mass down," you would be tempted to make a statement like, "If I put two point masses down, then I can combine the solution for mass #1 and mass #2 and get another solution: the solution for both masses combined.".

That is true, but only if you have a linear theory.

Newtonian gravity is a linear theory: the gravitational field is the gravitational field of every object added together and superimposed atop one another.

But Einstein's equations are nonlinear, which means you cannot do that. If you know the spacetime curvature for a single point mass, and then you put down a second point mass and ask, "How is spacetime curved now?" we cannot write down an exact solution. In fact, even today, more than 100 years after general relativity was first put forth, there are still only about ~20 exact solutions known in relativity, and a spacetime with two point masses in it still is not one of them..

A photo of Ethan Siegel at the American Astronomical Society's hyperwall in 2017, along with the first Friedmann equation at right — what is occasionally known as the most important equation in the universe and one of the rare exact solutions in general relativity. Credit: Harley Thronson / Perimeter Institute).

Originally, Einstein formulated general relativity with only the first and last terms in the equations, that is, with the Einstein tensor on one side and the stress-energy tensor (multiplied by the Einstein gravitational constant) on the other side.

A cosmological constant, mathematically, is literally the only "extra" thing you can add into general relativity without fundamentally changing the nature of the relationship between matter and energy and the curvature of spacetime.

The heart of general relativity, however, is not the cosmological constant, which is simply one particular type of "energy" you can add in but rather the other two more general terms.

The Einstein tensor, Gμν, tells us what the curvature of space is, and it is related to the stress-energy tensor, Tμν, which tells us how the matter and energy within the universe is distributed.

Quantum gravity tries to combine Einstein's General theory of Relativity with quantum mechanics.

If we ignored 15 out of the 16 Einstein equations and simply kept the "energy" component, you would recover the theory it superseded: Newton's law of gravitation.

But you are also allowed to put in any distribution of matter and energy, as well as any collection of fields and particles that you like, and if you can write it down, Einstein's equations will relate the geometry of your spacetime to how the universe itself is curved to the stress-energy tensor, which is the distribution of energy, momentum, and stress.

Co-infection can create a predicament for viruses when you consider that they need to compete for the same resource: you

Some viruses appear to block other viruses, while some viruses seem to like each other

The University of Glasgow study investigated what happens when you infect cells in a dish with two human respiratory viruses

They found that some of the human lung cells in the dish contained both viruses

And, by looking closely at those co-infected cells, they found that the viruses that were emerging from the cell had structural characteristics of both IAV and RSV

The new “chimeric" virus particles had proteins of both viruses on their surface and some even contained genes from the other

This is the first evidence of this occurring from co-infection of distinct respiratory viruses

Follow-up experiments in the same paper showed that these new chimeric viruses were fully functional and could even infect cells that were rendered resistant to influenza, presumably gaining access using the RSV proteins could even get into a broader range of human cells than either virus alone could

It's important to point out that the researchers in this study did not perform any genetic engineering between two viruses and only modelled what is already happening in the real world, but using safer laboratory strains of viruses under lab conditions

We know about the significant role co-infection can play in a virus's life, such as during influenza antigenic shift or the curious case of hepatitis D virus borrowing bits of the other viruses, such as hepatitis B, to spread

Together, this work shows the complex and often messy interactions between viruses during the winter

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