EINSTEIN FIELD

1. EINSTEIN FIELD EQUATIONS

G E N E R A L T H E O RY O F G R A V I T Y

2. Einstein field equations.

EINSTEIN FIELD EQUATIONS.
GEOMETRY OF SPACETIME =
ENERGY AND MATTER
Left-hand side: curvature
Right-hand side: source
• Rμν: Ricci tensor
• Tμν: stress-energy tensor
contains energy density
momentum, pressure and stress
• R: scalar curvature
• gμν: metric tensor
• Λ: cosmological constant
• 8πG/c⁴ fixes the Newtonian limit
Next, we will consider
Einstein’s equations for a
vacuum.

3. Spacetime interval.

SPACETIME INTERVAL.
If x⁰ = ct, then the interval as can also be written
Let’s introduce gμν:

4. Lorentz metric tensor.

LORENTZ METRIC TENSOR.
• The metric tells us how to measure distance and time in spacetime.
• It is the main unknown of general relativity.
• In flat spacetime, the metric becomes the Minkowski (Lorentz) metric.
• In a gravitational field, gμν depends on position.

5. COVARIANT Differentiation.

COVARIANT DIFFERENTIATION.
An ordinary derivative is not enough in curved spacetime, because the local basis also
changes from point to point.
VECTOR
COVECTOR
• ∂ν describes the change of components.
• The Christoffel symbols Γ encode the change of the basis.
• Covariant derivatives preserve tensor character.
IDEA: covariant derivative = change of components + change of basis

6. RIEMANN curvature tensor.

RIEMANN CURVATURE TENSOR.
The Riemann tensor measures intrinsic curvature. It tells us how much a vector
changes after parallel transport around a small closed loop.
• If spacetime is flat, then the Riemann tensor
vanishes.
• It is the full local description of curvature.
• Physically, it is related to tidal gravity.

7. Ricci tensor. scalar curvature.

RICCI TENSOR. SCALAR CURVATURE.
The Ricci tensor is a contraction of the Riemann tensor. It describes how a small
cloud of nearby geodesics (a straight line in curved space) focuses or expands.
Interpretation of the Ricci tensor:
• Tracks volume change of a small freely
falling cloud
• Focuses or defocuses nearby geodesics
• Is the part of curvature directly linked to
matter
Interpretation of scalar curvature:
• Shows the total curvature of space
at a point.

8. VACUUM EQUATIONS.

In vacuum there is no matter, so Tμν = 0. If we also take Λ = 0, the Einstein equations
simplify to the vacuum equations.

9. conclusion.

CONCLUSION.
Important wairning: Rμν = 0 does NOT mean flat spacetime. The full Riemann tensor may still
be non-zero.
Example: outside a star or a black hole, spacetime is vacuum but still curved.
KEY TAKEAWAYS:
• The metric defines geometry.
• Covariant derivatives make tensor equations meaningful.
• Riemann curvature measures tidal gravity.
• Einstein equations connect curvature to matter.

10. THE Gravitational END

THE GRAVITATIONAL END