Einstein Field Equations
The fundamental equations of General Relativity that describe how matter and energy curve spacetime, and how curved spacetime tells matter how to move.
Published by Albert Einstein in 1915 | Foundation of General Relativity
What Does This Equation Mean?
"Matter tells spacetime how to curve, curved spacetime tells matter how to move."
Left Side: Gμν
The Einstein tensor describes the curvature of spacetime. This is the geometric side - how spacetime bends and warps.
Right Side: Tμν
The stress-energy tensor describes matter and energy. This includes mass, momentum, pressure, and stress.
The Equation
The equals sign connects them: geometry = matter. Mass and energy create curvature, and curvature affects motion.
Visual Understanding: Spacetime Curvature
The Einstein Field Equations describe how mass and energy warp the fabric of spacetime:
The grid represents spacetime. Mass causes curvature, and objects follow "straight lines" (geodesics) in this curved space.
Key Concepts to Understand
1. Spacetime Curvature vs. Newtonian Gravity
| Concept | Newtonian Gravity | General Relativity (Einstein) |
|---|---|---|
| What is gravity? | A force between masses | Curvature of spacetime |
| Space and time | Separate, absolute | Unified as spacetime, dynamic |
| How objects move | Force F = ma | Follow geodesics (straight lines in curved space) |
| Equation | F = Gm₁m₂/r² | Gμν = 8πG Tμν |
| Valid when | Weak fields, slow speeds | Always (includes Newtonian limit) |
2. The Metric Tensor: Measuring Distances
The metric tensor gμν is the fundamental object that tells you how to measure distances in curved spacetime:
Example - Schwarzschild metric (around a spherical mass):
3. The Einstein Tensor: Encoding Curvature
The Einstein tensor is constructed to be:
- Symmetric: Gμν = Gνμ
- Divergence-free: ∇μGμν = 0 (ensures energy-momentum conservation)
- Built from curvature: Gμν = Rμν - ½Rgμν
The Ricci tensor Rμν and Ricci scalar R are built from the Riemann curvature tensor, which measures how vectors change when parallel transported around loops.
📚 Learning Resources
🎥 YouTube Video Explanations
Einstein Field Equations - PBS Space Time
Excellent introduction to the field equations and their meaning.
Watch on YouTube → 16 minGeneral Relativity Explained - Veritasium
Intuitive visual explanation of curved spacetime and geodesics.
Watch on YouTube → 15 minVisualizing Einstein's Field Equations - ScienceClic
Beautiful animations showing how curvature works.
Watch on YouTube → 9 minGeneral Relativity Course - Leonard Susskind
Full lecture series from Stanford (for serious learners).
Watch Playlist → 12 lectures📖 Articles & Textbooks
- Wikipedia: Einstein Field Equations | General Relativity | Introduction to GR
- Original Paper (1915): Einstein, A. "The Field Equations of Gravitation" [English translation]
- Textbook (Beginner): "Spacetime and Geometry" by Sean Carroll [Book website]
- Textbook (Advanced): "Gravitation" by Misner, Thorne, Wheeler (MTW) [Wikipedia]
- Online Course: MIT OpenCourseWare - General Relativity (8.962) [Free lectures & notes]
🔧 Interactive Tools
- Einstein's Field Equations Calculator: Wolfram Alpha
- Geodesic Simulator: Schwarzschild Spacetime Simulator
- Black Hole Ray Tracer: Starless (WebGL black hole visualization)
📝 Key Terms & Concepts
Geodesic
The "straightest possible" path through curved spacetime. Objects in free fall follow geodesics.
Learn more →Diffeomorphism
A smooth, invertible coordinate transformation. Einstein's equations are diffeomorphism invariant (coordinate independent).
Learn more →Covariant Derivative
The generalization of ordinary derivatives to curved spacetime. Accounts for how basis vectors change.
Learn more →Riemann Tensor
The fundamental measure of spacetime curvature. A rank-4 tensor with 20 independent components in 4D.
Learn more →Christoffel Symbols
Connection coefficients Γλμν that describe how basis vectors change. Not tensors themselves.
Learn more →Equivalence Principle
Locally, gravity is indistinguishable from acceleration. Foundation of Einstein's geometric view.
Learn more →🔬 Experimental Verification
The Einstein Field Equations have been tested to extraordinary precision:
✓ Mercury's Perihelion Precession (1915)
Einstein's first major success. Explained the 43 arcseconds/century discrepancy in Mercury's orbit.
✓ Gravitational Lensing (1919)
Eddington's solar eclipse expedition confirmed light bending around the Sun, making Einstein world-famous.
✓ Gravitational Redshift (1959)
Pound-Rebka experiment confirmed that clocks run slower in gravitational fields.
✓ Binary Pulsar Decay (1974)
Hulse-Taylor binary pulsar confirms gravitational wave emission. Nobel Prize 1993.
✓ Gravitational Waves (2015)
LIGO directly detected gravitational waves from black hole mergers. Nobel Prize 2017.
✓ Black Hole Shadow (2019)
Event Horizon Telescope imaged the shadow of M87* black hole, confirming predictions.
Connection to Principia Metaphysica
Principia Metaphysica extends Einstein's General Relativity to higher dimensions in the 2T physics framework:
26D Bulk → 13D Shadow → 6D → 4D
The Einstein Field Equations generalize to each dimensional stage:
- 26D bulk (24,2): GAB + Λ26gAB = 8πG26 TAB with signature (24,2)
- 13D shadow (12,1): After Sp(2,R) gauge fixing: GMN + Λ13gMN = 8πG13 TMN
- 6D bulk (5,1): After G₂ compactification with flux-dressed geometry
- 4D observed (3,1): The standard Einstein equations we observe
The Einstein-Hilbert Action shows how these equations arise from a variational principle at each dimensional stage.
💡 Practice Problems
Test your understanding with these exercises:
Problem 1: Newtonian Limit
Show that in the weak-field, slow-velocity limit, the Einstein Field Equations reduce to Newton's law of gravity: ∇²Φ = 4πGρ
Hint
Start with gμν = ημν + hμν where |hμν| << 1. Keep only first-order terms in h.
Problem 2: Schwarzschild Radius
Calculate the Schwarzschild radius rs = 2GM/c² for: (a) Earth, (b) Sun, (c) a stellar-mass black hole (10 M☉)
Solution
(a) Earth: rs ≈ 9 mm
(b) Sun: rs ≈ 3 km
(c) 10 M☉: rs ≈ 30 km
Problem 3: Energy Conservation
Prove that the Einstein tensor is divergence-free: ∇μGμν = 0. Why does this guarantee energy-momentum conservation?
Hint
Use the Bianchi identity: ∇μRμνρσ + cyclic permutations = 0
Where Einstein Field Equations Are Used in PM
This foundational physics appears in the following sections of Principia Metaphysica:
Where Einstein Field Equations Are Used in PM
This foundational physics appears in the following sections of Principia Metaphysica: