Einstein’s Field Equations

Question 2

What is the Einstein Field Equation (EFE)?

Answer 2

G_{μν} + Λg_{μν} = 8πG T_{μν}.

Question 3

What does G_{μν} represent?

Answer 3

Einstein tensor: G_{μν} = R_{μν} - 1/2 g_{μν}R.

Question 4

What does T_{μν} represent?

Answer 4

The energy-momentum tensor of matter and fields.

Question 5

What does Λg_{μν} represent?

Answer 5

Cosmological constant term, acts like vacuum energy.

Question 6

What is the significance of the Einstein tensor?

Answer 6

It encodes the curvature of spacetime due to mass-energy.

Question 7

What is the physical meaning of EFE?

Answer 7

Spacetime curvature (LHS) = matter content (RHS).

Question 8

What constant appears in the EFE?

Answer 8

Newton’s constant G and π (in natural units, 8πG).

Question 9

What is the role of the cosmological constant Λ?

Answer 9

Can accelerate the universe’s expansion, models dark energy.

Question 10

What are the units of T_{μν}?

Answer 10

Energy density (e.g., kg m⁻¹ s⁻² or J/m³).

Question 11

What equation ensures energy-momentum conservation?

Answer 11

∇^μ T_{μν} = 0.

Question 12

How many independent equations are in the EFE?

Answer 12

10 equations due to the symmetry of G_{μν} and T_{μν}.

Question 13

Why are the Einstein equations nonlinear?

Answer 13

Because G_{μν} depends nonlinearly on the metric and its derivatives.

Question 14

What is a vacuum solution to EFE?

Answer 14

G_{μν} = 0 or R_{μν} = 0 when T_{μν} = 0.

Question 15

What are examples of vacuum solutions?

Answer 15

Schwarzschild, Kerr, and Minkowski spacetimes.

Question 16

How does one solve the Einstein equations?

Answer 16

Assume a symmetric form for the metric, compute G_{μν}, solve for g_{μν}.

Question 17

What is the trace of the Einstein equation?

Answer 17

R - 4Λ = -8πG T (trace of energy-momentum tensor).

Question 18

What is a perfect fluid energy-momentum tensor?

Answer 18

T_{μν} = (ρ + p)u_μ u_ν + p g_{μν}.

Question 19

What symmetries simplify solving EFE?

Answer 19

Spherical, axial, planar, and cosmological symmetries.

Question 20

What is the Einstein-Hilbert action?

Answer 20

S = ∫ (R - 2Λ)√|g| d⁴x + S_matter, variational principle for EFE.

Question 21

What happens to EFE in weak fields?

Answer 21

They reduce to Poisson’s equation: ∇²Φ = 4πGρ.

Question 22

What is the Newtonian limit of EFE?

Answer 22

When the metric is close to flat and velocities are small.

Question 23

Why is T_{μν} symmetric?

Answer 23

Due to conservation laws and Noether’s theorem for translations.

Question 24

What is meant by the ‘left-hand side’ of EFE?

Answer 24

Geometric side: curvature quantities (G_{μν} + Λg_{μν}).

Question 25

What is meant by the 'right-hand side' of EFE?

Answer 25

Matter/energy content: 8πG T_{μν}.

Question 26

What is the weak energy condition?

Answer 26

T_{μν}u^μu^ν ≥ 0 for all timelike vectors u^μ.

Question 27

What is the strong energy condition?

Answer 27

(T_{μν} - 1/2 g_{μν} T)u^μu^ν ≥ 0.

Question 28

What is the dominant energy condition?

Answer 28

T_{μν}u^μ is a future-pointing timelike or null vector.

Question 29

What is the null energy condition?

Answer 29

T_{μν}k^μk^ν ≥ 0 for all null vectors k^μ.

Question 30

What is the vacuum Einstein equation?

Answer 30

G_{μν} = 0 or R_{μν} = 0.

Question 31

What is the Schwarzschild solution?

Answer 31

Solution to EFE in vacuum with spherical symmetry: black holes.

Question 32

What role does symmetry play in solving EFE?

Answer 32

It reduces the number of independent functions in the metric.

Question 33

What is the energy-momentum tensor for electromagnetism?

Answer 33

T_{μν} = F_{μα}F^α_ν - 1/4 g_{μν} F_{αβ}F^{αβ}.

Question 34

Can EFE describe gravitational waves?

Answer 34

Yes, small perturbations of the metric satisfy wave equations.

Question 35

What is a stress-energy source?

Answer 35

Any field or fluid with energy density and pressure contributing to T_{μν}.

Question 36

What does it mean to 'solve' the Einstein equations?

Answer 36

Find a metric g_{μν} that satisfies EFE for a given T_{μν}.

Question 37

What is the significance of √|g| in EFE?

Answer 37

It appears in the action and volume elements in curved spacetime.

Question 38

What is an isotropic and homogeneous solution?

Answer 38

FLRW metric — used in cosmology.

Question 39

Why is EFE a second-order differential equation?

Answer 39

Because it involves up to second derivatives of the metric.

Question 40

Can you superpose solutions in EFE?

Answer 40

No, due to the nonlinearity of the equations.

Question 41

What are the boundary conditions for EFE?

Answer 41

Asymptotic flatness or behavior at infinity or horizon (depending on context).