Chapter 6: Path to General Relativity
Special relativity handles inertial frames; general relativity extends to accelerated frames and gravity. Einstein's key insight—the equivalence principle—shows that gravity curves spacetime itself.
Limitations of Special Relativity
- • Only applies to inertial (non-accelerating) frames
- • Cannot handle gravity (which accelerates everything equally)
- • Uses flat Minkowski spacetime (ημν)
- • Incompatible with Newtonian gravity (instantaneous action at a distance)
The Equivalence Principle
"A freely falling observer feels no gravity—they're in an inertial frame."
Einstein's insight: Locally, gravity is indistinguishable from acceleration. A person in a falling elevator can't tell if they're falling in a gravitational field or floating in empty space.
Falling Elevator
Inside: appears like zero gravity. Objects float. Light travels straight.
Accelerating Rocket
Inside: feels like gravity. Objects fall. Light bends (!)
Gravity as Curved Spacetime
In GR, mass-energy curves spacetime. The metric tensor gμν(x) replaces ημν:
\( ds^2 = g_{\mu\nu}(x) dx^\mu dx^\nu \)
Free particles follow geodesics—the "straightest possible" paths in curved spacetime. Gravity isn't a force; it's the curvature of spacetime telling matter how to move.
Preview: Einstein Field Equations
\( G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \)
"Matter tells spacetime how to curve; spacetime tells matter how to move." These equations describe black holes, gravitational waves, the expanding universe, and more.
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Special Relativity: Course Summary
• The speed of light is constant in all inertial frames (c ≈ 3×10⁸ m/s)
• Space and time are unified into 4D spacetime
• Moving clocks run slow; moving rulers contract
• Simultaneity is relative—"now" depends on the observer
• Energy and mass are equivalent: E = mc²
• Four-vectors and tensors transform covariantly under Lorentz transformations
• Electromagnetism is naturally relativistic; E and B fields unify into Fμν
Congratulations! You've completed the Special Relativity course. The journey continues with General Relativity, Quantum Field Theory, and beyond.