How to Understand Einstein's General Relativity Equation

Everyone is amazed by the idea of gravitation explained as a curvature of spacetime. Einsteins one of the biggest achievement was theory of relativity which almost changed the whole concept of gravitation. Many wants to understand his equation of general relativity but look for easier explanations. So we'll try our best to understand it in simplest way possible.

• Although the Einstein field equations seem quite straightforward, they actually include a great deal of complexity.
• A seemingly simple equation that links the curvature of spacetime to the universe's matter and energy is really composed of 16 intricate equations.
• It demonstrates how gravity differs fundamentally from all other forces while also being the only one we can generally understand.

Although Einstein is a legendary figure in science for many reasons, including  E = mc², the photoelectric effect, and the idea that the speed of light is constant for all, his theory of gravitation, general relativity, is his most important and least understood contribution. Prior to the discovery of gravity by Albert Einstein, we understood gravitation in terms of Newtonian physics, which states that everything in the universe that has a mass instantly attracts every other object that has a mass, depending on the magnitude of their masses, the gravitational constant, and the square of the distance between them.  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 notion, according to which matter and energy tell spacetime to curve and spacetime tells matter and energy to move, constituted a revolutionary new understanding of the cosmos. General relativity has passed every observational and experimental test we have ever devised. It was proposed in 1915 by Einstein and confirmed 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. However, despite its effectiveness over more than a century, hardly nobody is aware of the true nature of the one equation that underlies general relativity. Here is what it really implies in plain English.

Photo Credit - University of Tokyo; Kavli IPMU

Since there are just a few symbols in this equation, it appears to be quite straightforward. But it's extremely intricate.

• The first Gμν is known as the Einstein tensor and represents the curvature of space.
• The second, Λ, is the cosmological constant, a positive or negative quantity of energy inherent in the structure of space itself.
• The third term gμν is known as the metric and mathematically encodes the properties of each point in spacetime.
• The fourth term, 8πG/c^4, is simply a product of constants, known as Einstein's gravitational constant, which corresponds to Newton's gravitational constant (G).
• The fifth term Tμν is known as the stress-energy tensor and represents the local (nearby) energy, momentum and stress in this spacetime.

In order to connect the geometry of spacetime to all the matter and energy contained in it, we just need these five terms, which are all connected to one another by what are known as the Einstein field equations. This is the essence of general relativity.

Photo Credit - Vysotsky / Wikimedia Commons

You might be perplexed by the strange "μ ν " Greek letter subscripts that appear at the base of the Einstein tensor, the metric, and the stress-energy tensor. The majority of the time, when we write down an equation, it is a scalar equation, which is an equation that only reflects one equivalence, where the total of the left and right sides equals the right and left, respectively. But we may also describe systems of equations with a single straightforward formulation that perfectly sums up these relationships.

E = mc² is a scalar equation. This is because energy (E), mass (m), and speed of light (c) all have a single unique value. But Newton's F=ma is not a single equation, but three separate equations. Fx = max in the "x" direction, Fy = may in the "y" direction, Fz = maz in the "z" direction. The fact that in general relativity there are four dimensions (three in space and one in time), and two subscripts that physicists know as indices, means that there is not one equation, nor even three or four.  Instead, each of the four dimensions (t, x, y, z) affects each of the other four dimensions (t, x, y, z), for a total of 4 × 4 or 16 equations.

Why do we require so many equations to simply express gravitation, whereas Newton only required one?

Photo Credit - Christopher Vitale of Networkologies and The Pratt Institute

Because geometry is a complex subject, we are working in four dimensions, and because events in one dimension or even one location have the potential to spread throughout the universe if enough time is given. Our world has three space dimensions and one time dimension, which allows us to formally consider its geometry as a four-dimensional manifold.

In Riemannian geometry, the manifold does not have to be straight and rigid, it can be arbitrarily curved, so we can divide its curvature into two parts: the part that distorts the object's volume and the part that distorts the object's shape. The "Ricci" part distorts the volume. This is important  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 the shape distorting, which counter-intuitively plays no part in Einstein's field equations.

So the Einstein field equations are not a single equation, but a set of 16 different equations, one for each "4 × 4" combination. When a component or aspect of the Universe changes like Spatial curvature at any point or in any direction. All other components may change accordingly. This framework takes the concept of differential equations to the next level in many ways.

A differential equation is an equation where you can:

• You can specify initial conditions for the system, such as what is present, where, and how it is moving.
• We can then substitute these terms into the differential equation.
• And the equation tells us how these things will play out in the next moment.
• Put that information back into the differential equations and you'll know what happens next the next moment.

It's a very powerful framework, and that's exactly why Newton had to invent calculus to enable the scientific understanding of things like motion and gravity.

It is an incredibly potent framework, and it is the reason why calculus had to be developed by Newton in order for concepts like motion and gravitation to be understood on a scientific level.

However, when dealing with general relativity, there isn't simply one equation, or even a group of independent equations, which propagates and develops in its own dimension. Instead, we have 16 connected, interdependent equations because what happens in one direction or dimension affects all the others. As things travel and accelerate through spacetime, the stress-energy and spatial curvature change.

These "16 equations" are not, however, truly original. First of all, the Einstein tensor is symmetric, which implies that each component has a connection that relates one direction to the other. In particular, if your four time and space coordinates are (t, x, y, z), then:

• the “tx” component will be equivalent to the “xt” component,
• the “ty” component will be equivalent to the “yt” component,
• the “tz” component will be equivalent to the “zt” component,
• the “yx” component will be equivalent to the “xy” component,
• the “zx” component will be equivalent to the “xz” component,
• and the “zy” component will be equivalent to the “yz” component.

All of a sudden, there aren’t 16 unique equations but only 10.

The Bianchi Identities, a set of four connections, are what hold the curvature of these many dimensions together. Only six of the remaining 10 unique equations are independent due to the four connections, which further reduce the number of independent variables. This part's power gives us the freedom to choose any coordinate system we like, which is a literal manifestation of relativity's power: every observer, regardless of their position or motion, perceives the same physical laws, including the same general relativity rules.

There are further, very significant characteristics of these equations. The stress-energy tensor's divergence, in instance, always returns zero, not just overall but for each individual component. As a result, there are four symmetries: no divergence in any of the space dimensions or time dimensions, and whenever there is a symmetry in physics, there is also a conserved quantity.

In general relativity, those conserved quantities correspond to momentum in the x, y, and z directions as well as energy (for the time dimension) (for the spatial dimensions). Just like that, momentum and energy are conserved for individual systems, at least locally in your immediate area. Energy and momentum must always be conserved for any local system in general relativity; this is a necessity of the theory, despite the fact that it is impossible to describe concepts like "global energy" overall.

The fact that general relativity is a nonlinear theory distinguishes it from the majority of other physical theories. 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's true, but only if you have linear theory. Newtonian gravity is a linear theory. The gravitational field is the gravitational field of each object added and superimposed. So is Maxwell's electromagnetism,  the electromagnetic field of two charges, two currents, or a charge and a current can all be calculated individually and added together to give the net electromagnetic field. This also applies to quantum mechanics, since the Schrödinger equation (of the wavefunction) is also linear.

However, since Einstein's equations are nonlinear, you cannot accomplish that. We are unable to provide an exact answer if you know the spacetime curvature for a single point mass and then add a second point mass and ask, "How is spacetime curved now?" Even now, more than a century after general relativity was initially proposed, only roughly 20 perfect solutions to relativity are known, and a spacetime containing two point masses is still not one of them.

Originally, Einstein formulated general relativity using only the first and last terms of the equation, one is the Einstein tensor and the other is the stress energy tensor (multiplied by Einstein's gravitational constant). He simply added the cosmological constant because he could not withstand the consequences of the universe having to expand or contract.

And yet, even if nature turned out not to have a non-zero one (in the form of today's dark energy), the cosmological constant itself would have been a revolutionary addition for a straightforward but fascinating reason. Mathematically speaking cosmological constant is the only "additional" component you can introduce into general relativity without significantly altering the way matter and energy interact with one another and the curvature of spacetime.

However, the core of general relativity is not the cosmological constant, nor the specific kind of "energy" that can be added, but the other two more general terms. The Einstein tensor Gμν tells us what the curvature of space is. This is related to the stress-energy tensor Tμν, which describes how matter and energy are distributed in the universe.

Quantum gravity, theory which describes gravity in regimes where quantum effects cannot be ignored

In our universe, we almost always do approximations. Ignoring 15 of Einstein's 16 equations and simply leaving the "energy" factor alone restores an alternative theory to them: Newton's law of universal gravitation. If instead we make the universe symmetrical in all spatial dimensions and unrotated, we get an isotropic, uniform universe, governed by the Friedmann equation (and therefore must expand or contract). Indeed, on the grandest cosmic scale, this seems to describe the universe we live in.

However, if you can put together a collection of fields and particles as well as any distribution of matter and energy, Einstein's equations will relate the geometry of your spacetime to the curvature of the universe as a whole and to the stress-energy tensor, which is the distribution of energy, momentum, and stress. If you can do this, you can use any distribution of matter and energy as well.

The basic discrepancies between both notions, particularly the inherently nonlinear character of Einstein's theory, will need to be addressed if there is a true "theory of everything" that describes both gravity and the quantum universe. The unification of gravity with the other quantum forces is currently one of the most challenging goals in all of theoretical physics due to their radically different features.

There are so many questions that we still need to ask. Science develops day by day, month by month and years by years. We can hope that one day we'll able to find a way to explain combined behaviour or all fundamental forces in Universe. Theory of everything is what all we are looking for..

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