einstein field equation

1. Presentation on
Einstein Field Equations
and
Bianchi Identity
Presented by- Diksha Pandey (cutie)
Roll No - 2240130420010

2. Einstein Field Equations
The Einstein Field Equations refer to a set
of equations in physics that relate the
energy-momentum tensor of matter and
energy in the universe to the curvature
of space-time, incorporating the Newton
gravitational constant and the
cosmological constant.
Introduction
𝐺𝜇𝜈+ Λ 𝑔𝜇𝜈=
8 𝜋 𝐺
𝑐
4
𝑇 𝜇𝜈
𝐺 𝜇𝜈
𝑇 𝜇𝜈
𝑐
𝐺
Einstein Tensor
𝑔 𝜇𝜈
Metric Tensor
Stress – Energy Tensor
Speed of Light
Newton’s Gravitational constant
Λ Cosmological Constant

3. Einstein Tensor
Gμν=Rμν −
1
2
R gμν
The Einstein Tensor​
, is a fundamental component
of the Einstein Field Equations (EFE). It describes
how spacetime is curved due to the presence of
mass and energy.
In easy words, it contains information about the
curvature of spacetime. This allows us to
understand how the region of spacetime we
happen to be considering is curved and warped..
=
Ricci Tensor (describes how volume elements change in
curved spacetime)
= Metric Tensor (defines distances in spacetime)
R = Ricci Scalar (sum of curvature across all directions)

4. 𝐺𝜇𝜈+ Λ 𝑔𝜇𝜈=
8 𝜋 𝐺
𝑐
4
𝑇 𝜇𝜈
Cosmological Constant Metric Tensor
The Metric Tensor is a fundamental
concept in General Relativity. It
describes the geometry of spacetime
and determines the distances
between points in a curved
spacetime.
The cosmological constant is simply Einstein's
way to encode the fact that the universe is
expanding, into these equations. In other
words, the behavior of the spacetime fabric is
not only dependent on the mass / energy
inside it, but also on the fact that it just seems
to be inherently stretching.
In easy words , together they explain
the shape of the spacetime.
(Λ) (𝑔¿¿𝜇𝜈)¿

5. Stress – Energy Tensor
The Stress-Energy Tensor (also called the
Energy-Momentum Tensor) is denoted
by . It describes how matter and energy
are distributed in spacetime and how
they influence curvature (gravity).
This contains information about the distribution
of mass, energy, momentum, pressure, and so
on, within the region of spacetime we happen
to be considering. Often, this region happens to
be the entire universe.
In easy words, the Stress-Energy Tensor tells
spacetime how to curve.
𝜇𝜈+ Λ 𝑔𝜇𝜈=
8 𝜋 𝐺
𝑐
4
𝑇 𝜇𝜈

6. Significance of Einstein Field equations
Explains Gravity as Spacetime Curvature
1. Before EFE, Newtonian gravity described
gravity as a force between masses.
2. EFE showed that gravity is the bending of
spacetime caused by mass and energy.
Predicts and Describes Black Holes
1. The EFE led to solutions like the Schwarzschild
metric, which describes black holes.
2. It predicts event horizons, singularities, and
gravitational time dilation near black holes.
Explains the Expansion of the Universe
1. The Friedmann equations, derived from EFE,
describe how the universe expands.
2. EFE supports the Big Bang Theory and
predicts the role of dark energy in cosmic
expansion.
Guides Future Physics Theories
1. EFE remains the gold standard in gravitational
physics.
2. Scientists use it to search for quantum gravity and
understand phenomena like dark matter and dark
energy.

7. Bianchi Identity in General Relativity
The Bianchi Identity is a fundamental
mathematical property of spacetime curvature.
It ensures the consistency of Einstein’s Field
Equations (EFE) and guarantees the
conservation of energy and momentum in
General Relativity.
The Bianchi Identity is written as:
​
=0
where:
• is the covariant derivative (which
ensures the equation is valid in curved
spacetime).
• is the Einstein Tensor (which
describes spacetime curvature)
This equation tells us that the Einstein Tensor is
divergence-free—a crucial feature for the
consistency of General Relativity.

8. Why is the Bianchi Identity Important?
1. Conservation of Energy and Momentum:
The identity ​
=0
• ensures that energy and momentum are always
conserved in General Relativity.
• This aligns with the physical law that energy cannot be
created or destroyed.
2. Ensures the Validity of Einstein’s Equations:
• Since is derived from the Ricci Tensor the Bianchi
Identity also applies to the curvature of spacetime.
• This makes sure that any solution to EFE is physically
consistent

9. Connection to Einstein’s Equations
Since the Einstein Tensor is related to the
Stress-Energy Tensor ​via EFE:
𝑮𝝁𝝂 =
𝟖 𝝅 𝑮
𝒄
𝟒
𝑻 𝝁𝝂
Applying the Bianchi Identity to this
equation, we get:
​
=0
which confirms that energy and momentum are
conserved in curved spacetime—a crucial
aspect of General Relativity.
Conclusion
The Bianchi Identity is a fundamental
property of spacetime curvature that
ensures energy-momentum conservation
and guarantees that Einstein’s Field
Equations remain consistent with physical
laws. It acts as a bridge between the
mathematical structure of spacetime and
the physical behavior of energy and
matter.

10. Thank you