13.7 Einstein's Theory of Gravity

Einstein's theory of gravity revolutionized our understanding of the universe. It describes gravity not as a force, but as a consequence of spacetime curvature caused by massive objects. This groundbreaking concept explains phenomena like planetary orbits and light bending near massive bodies.

The theory introduces the principle of equivalence, which states that gravity's effects are indistinguishable from acceleration. This idea leads to fascinating implications, such as the prediction of black holes and gravitational waves, which have been confirmed through observations and experiments.

Einstein's Theory of Gravity

Einstein's general theory of relativity

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• Geometric theory of gravity describes gravity as consequence of spacetime curvature not a force
• Massive objects curve spacetime and this curvature influences motion of other objects (planets orbiting the Sun)
• Spacetime is four-dimensional continuum with three spatial dimensions and one time dimension
• Presence of mass or energy curves spacetime
  • More massive objects induce greater curvature of spacetime
• Objects in free fall follow geodesics (straight paths) in curved spacetime
  • Paths appear curved from outside perspective due to spacetime curvature (light bending near the Sun)

Principle of equivalence in gravity

• Effects of gravity indistinguishable from effects of acceleration
  • Observer in closed elevator cannot distinguish between being stationary in gravitational field and being accelerated in absence of gravity (rocket accelerating in space)
• Inertial mass and gravitational mass are equivalent
  • Inertial mass measures object's resistance to acceleration
  • Gravitational mass determines strength of object's gravitational field and response to external gravitational fields
• All objects fall at the same rate in a gravitational field regardless of mass or composition
  • Confirmed experimentally to high degree of accuracy (feather and hammer drop on the Moon)

Calculation of Schwarzschild radius

• Radius of event horizon for non-rotating black hole
  • Boundary beyond which nothing including light can escape black hole's gravitational pull (point of no return)
• Schwarzschild radius ($r_s$) given by formula $r_s = \frac{2GM}{c^2}$
  • $G$ gravitational constant $6.67 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}$
  • $M$ mass of object
  • $c$ speed of light $3.00 \times 10^8 \text{ m/s}$
• Sun (mass $\approx 1.99 \times 10^{30} \text{ kg}$) has Schwarzschild radius about 2.95 km
• Schwarzschild radius proportional to object's mass
  • More massive objects have larger Schwarzschild radii (supermassive black hole at Milky Way center)

Observational evidence for black holes

• Black holes emit no light so cannot be directly observed
  • Existence inferred from gravitational effects on nearby matter and light (stars orbiting galactic centers)
• Accretion disks around black holes
  • Matter falling into black hole forms hot bright accretion disk
  • X-ray emission from accretion disks indicates presence of compact massive objects (Cygnus X-1)
• Gravitational lensing
  • Black holes bend light due to strong gravitational fields causing distortions in images of background objects
  • Observations of gravitational lensing provide evidence for black holes (quasar lensing)
• Gravitational waves
  • Merger of two black holes produces gravitational waves (ripples in spacetime) predicted by general relativity
  • Detection of gravitational waves from binary black hole mergers by LIGO and Virgo confirms existence of black holes
• Supermassive black holes at galaxy centers
  • Observations of high-velocity stars orbiting Milky Way center suggest presence of supermassive black hole (Sagittarius A*)
  • Similar evidence found in other galaxies indicates supermassive black holes are common at galactic centers (M87*)

Mathematical foundations of general relativity

• Special relativity forms the basis for general relativity, extending its principles to non-inertial reference frames
• Tensor mathematics provides the mathematical framework for describing curved spacetime
• Riemann geometry is used to describe the curvature of spacetime in general relativity
• The energy-momentum tensor represents the distribution of mass and energy in spacetime, determining its curvature