[Physics Class Notes] on Einstein Field Equation Pdf for Exam

What is Einstein Field Equation?

The Einstein Field Equation is also known as Einstein’s equation. It is a set of ten equations that are extracted from the General Theory of Relativity, by Albert Einstein. Einstein’s equation describes the interaction of gravitation. Initially published in 1915, it is also widely called a tensor equation in the field of physics. It is actually the combination of ten different equations that are contained in the tensor equation. It describes gravity as a result of the space-time that is curved by both mass and energy. It is determined by the curvature of time and space at a particular point of time and space. It is also equated with momentum and energy at that point. The solutions of these equations are the components of the metric tensor that specifies the space-time geometry. The trajectories of these particles can then be found by the geodesic equations. 

 

How Accurate is the Einstein Field Equation of Gravity?

The Einstein Field Equation is not accurate. Even though the theory and the set of equations have passed the rest, they are incompatible with the known quantum theory. The latter also passed the experimental test, but unlike the further, it doesn’t have specifications. The problem with the Einstein Field Equation is that it needs the momentum and the energy to be precise at every point of space-time. As a result, it contradicts the uncertainty principle for all the quantum states. Despite being a problem at low energies and longer distances, it is also a conceptual incompatibility at every lab.

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Einstein Field Equation, General Relativity, and Cosmological Constant

Albert Einstein outlined the general theory of relativity back in 1915, and it is officially published in the next year. He stated it in an equation, which is a summary of ten equations. The equations completely changed how we shall understand the nature and the evolution of the universe. It presents us with the new idea that is fundamental to the basic fabric of reality, with respect to space-time. It is also malleable in nature. General relativity provides a picture of how gravity works, instead of exerting a pull on the massive objects that distort through space and time.  

Due to the origin of the equation, the constant in the Einstein Field Equation is known as Cosmological Constant. Back in 1931, it was highlighted by Hubble to Einstein that the universe is not constant; it is indeed expanding. Einstein, on considering the expansion, has introduced the cosmological constant while referring to it as the ‘biggest blunder of his life’. The universe is thought to have a residual constant, which is accelerating the expansion of the universe. 

 

Einstein Field Equation Formula

Gμυ+gμυΛ=8πGc4Tμυ

Where

• Gμ? is the Einstein tensor = Rμ?-½ Rgμ?

• Rμ? is the Ricci curvature tensor

• R is the scalar curvature

• gμ? is the metric tensor

• ? is a cosmological constant

• G is Newton’s gravitational constant

• c is the speed of light

• Tμ? is the stress-energy tensor

 

Einstein Field Equations’ Derivation

Einstein wants to explain that source of gravity = measure of curvature. The source of gravity is a stress-energy tensor, as given below: 

Tαβ = [begin{bmatrix} rho & 0 & 0 &0 \ 0 & P & 0 & 0\ 0& 0 & P &0 \ 0 & 0 &0 & P end{bmatrix}] —> [begin{bmatrix} rho & 0 & 0 &0 \ 0 & 0 & 0 & 0\ 0& 0 & 0 &0 \ 0 & 0 &0 & 0 end{bmatrix}]

In the matrix, we see that P is tending to become zero. But for Newton’s gravity, the mass density is a source of gravity. 

[frac{du^{i}}{dr}] + Γivauvuα= 0

[frac{du^{i}}{dr}] + Γi00== 0

[frac{du^{i}}{dr}] + [frac{1}{2}] [frac{partial g_{00}}{partial x^{i}}] = 0

[frac{du^{i}}{dr}] + [frac{1}{2}] [frac{partial phi }{partial x^{i}}] = 0

g00 = -(1 + 2ø)

But we know that ⛛2ø = 4πG?

Therefore,

R?v = -8πGT?v

where -8πGT?v is the constant.

 

What is an Einstein Tensor? 

Einstein tensor is Ricci tensor, which is trace-reversed. Using the Einstein Field Equation, it is used to describe space-time curvature that is in alignment with the conservation of momentum and energy. 

Einstein Tensor is defined as: 

G = R-½ gR

Where, 

 

What is a Stress-Energy Tensor?

Stress energy tensor, defined as the tensor Tαβ is called as a symmetrical tensor. It is used for describing the momentum density and energy of the gravitational field. It is given as: 

Tαβ = Tβα

 

Einstein Field Equation Open and Closed Models 

In the cosmological application of the general theory of relativity, it is assumed that the distribution of matter in space is isotropic and homogeneous as well. Because of this, the spatial geometry is also isotropic and homogeneous. Under this assumption, Einstein’s field equation led to two different non-static universe models: an open model with an infinite space of negative curvature and a closed model with a finite space of positive curvature.