Before we dive into this topic we should learn about Einstein Field Equation.
The Einstein Field Equation, which describes the gravity as a curvature of spacetime
1. Rᵤᵥ - ½Rgᵤᵥ + Λgᵤᵥ = 8πGTᵤᵥ /c⁴
Here
Rᵤᵥ is Ricci tensor which comes from Riemann tensor after, contracting one of the indices.
gᵤᵥ is metric tensor thus defines the surface or geometry
R is ricchi scalar which is trace of Rᵤᵥ
Tᵤᵥ is Stress Energy Tensor (Not Cauzy Stress Tensor)
G is universal gravitational constant(the same one used in Newtonian Gravity)
c is speed of light
Λ is cosmological constant
Here left-hand side is the curvature of spacetime, or gravity and right-hand side is the source term, like charge is the source of electric field and charge causes the curvature in electric field.
But let me write few more equations just to justify that metric tensor is enough to know everything about spacetime.
We use " metric tensor " to define Christoffel symbol
2. Γᵏᵢⱼ=½gᵏˡ(∂ᵢgⱼₗ+∂ⱼgᵢₗ−∂ₗgᵢⱼ)
We use Christoffel symbol to define Riemann Curvature Tensor
3. Rᵏₗₘₙ = ∂ₗΓᵏₘₙ - ∂ₙΓᵏₘₗ + ΓᵏₗₐΓᵃₘₙ - ΓᵏₙₐΓᵃₘₗ
Then, we use Riemann Curvature Tensor to define Ricci Tensor
4. Rᵏₗₖₘ= Rₗₘ
Finally, use it to define Ricci Scalar
5. Rᵤᵤ = R
Enough of that, as you are now familiar with Einstein Field Equation now we can talk some Physics. If we assume that Λ is "0" then, the equation turns to
Rᵤᵥ - ½Rgᵤᵥ = 8πGTᵤᵥ /c⁴
Which is fully attractive in nature, just like Newtonian Gravity is and under low speed and weak gravitation assumption the equation written above turns to
∇²φ = 4πGρ
Which is Poisson's law and g = - ∇φ
Now, we know this much, that General Relativity converges to Newtonian one, under two assumptions, but now let's talk about the Λ term in Einstein Field Equation a.k.a. Cosmological Constant. The sign and value of this term decides the fate of spacetime (that's why physicist often says gravitation decides the fate of our universe) if this term is negative, then in the sense that it could cancel out(as it will add with ricci scalar) and could cause gravitation to vanish and could make universe static. But it turns out the value of cosmological constant is not negative but positive it would reduce the value of ricci tensor. In the expression every term is dynamic(in the sense that, it depends on the scenario) but cosmological constant ain't and that's good as it is responsible for Big Bang. Let me elaborate it a bit, but for that you would have to accept one thing that if the positive stress tensor makes gravity then, negative stress tensor would make repulsive gravity.
Now, let's modify the equation a bit and assume Tᵤᵥ = 0.
The equation becomes
Rᵤᵥ - ½Rgᵤᵥ + Λgᵤᵥ = 0
Few modifications
Rᵤᵥ - ½Rgᵤᵥ = -Λgᵤᵥ
Now, here Λgᵤᵥ , term act as pseudo stress tensor as this can also cause spacetime to curve, because the curvature depends only on "gᵤᵥ" which affects "Rᵤᵥ" but doesn't affect "Λgᵤᵥ".
So, if I assume, Λ < 0
Then, it would act as normal gravity source but if Λ > 0, then
As R.H.S. is -ve so L.H.S must also be, which means it acts opposite to attractive gravity as repulsive gravity.
Now, all of these stuff is clear how cosmological constant can change, the fate of gravity, we all know from Inflation Theory that total energy of universe is "0" hence that would make Tᵤᵥ = 0 . But this doesn't affect Λ and hence can cause repulsive gravity even without energy or matter.
Now, we know what is this cosmological constant and how it could drive the expansion of our universe. But let's talk about Dark Energy, this is the most mysterious form of energy, as it drives the expansion rate of our universe, its just a name given to cosmological constant (in a sense). I haven't told you the whole glory detail but that would make you board, so I am gonna just tell you, there is an equation known as Friedman Equations, whicj involves the cosmological constant and it describes the acceleration of universe on the basis of value of cosmological constant long but some other parameters.
Thanks
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