Nyimbi Odero

Series 01 · a numbered run · kept in public

Part 5: Quantum Fields in Curved Spacetime

Chapter: Lesson 4: Multivariable Calculus II

The Marriage of Quantum Fields and Gravity

When we place quantum fields in curved spacetime, we enter a realm where quantum mechanics meets gravity. This semi-classical approach, while not a complete theory of quantum gravity, reveals fascinating phenomena like Hawking radiation and provides crucial insights into how quantum fields behave in gravitational fields.

Let’s organize our exploration using the mnemonic “SPACETIME”:

Scalar fields in curved space Propagators and Green functions Adiabatic vacuum states Covariant quantization Energy-momentum tensor Time evolution Inequivalent vacua Mode expansions Energy conditions

Fields in Curved Spacetime

  1. Covariant Klein-Gordon Equation
(□−m2)ϕ=1−g∂μ(−ggμν∂νϕ)−m2ϕ=0(\square - m^2)\phi = \frac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}g^{\mu\nu}\partial_\nu\phi) - m^2\phi = 0

where:

  • g is the determinant of the metric

  • gμν is the spacetime metric

  • ∇μ is the covariant derivative

  1. Mode Expansion
ϕ^(x)=∑k\hat{\phi}(x) = \sum_k

a_k u_k(x) + a_k^\dagger u_k^*(x)

where:

  • uk(x) are solutions to Klein-Gordon equation

  • ak, ak† are creation/annihilation operators

  • Inner product defined by Klein-Gordon norm

Vacuum States and Particle Creation

  1. Klein-Gordon Inner Product
(f,g)=i∫Σ(f∗∂μg−g∂μf∗)nμhdn−1x(f,g) = i\int_\Sigma (f^*\partial_\mu g - g\partial_\mu f^*)n^\mu\sqrt{h}d^{n-1}x

where:

  • Σ is a Cauchy surface

  • nμ is its normal vector

  • h is the induced metric

  1. Bogoliubov Transformations Between two sets of modes:
uk=∑j(αkjvj+βkjvj∗)u_k = \sum_j (\alpha_{kj}v_j + \beta_{kj}v_j^*)

Leading to particle creation:

⟨>01∣N^∗k(2)∣01⟩=∑j∣β∗kj∣2\langle > 0_1|\hat{N}*k^{(2)}|0_1\rangle = \sum_j |\beta*{kj}|^2

Energy-Momentum Tensor

  1. Classical Expression
Tμν=∂μϕ∂νϕ−12gμνT_{\mu\nu} = \partial_\mu\phi\partial_\nu\phi - \frac{1}{2}g_{\mu\nu}

g^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi + m^2\phi^2

  1. Quantum Version Requires regularization:
⟨Tμν⟩=lim⁡x′→x\langle T_{\mu\nu}\rangle = \lim_{x'\to x}

\partial_\mu\partial_\nu’ G^{(1)}(x,x’) - …

where G⁽¹⁾ is the Hadamard Green function.

Hawking Radiation

  1. Temperature Formula For Schwarzschild black hole:
TH=ℏc38πGMkBT_H = \frac{\hbar c^3}{8\pi GM k_B}

Derived from mode analysis near horizon.

  1. Unruh Effect Accelerated observers see thermal bath:
TU=ℏa2πkBcT_U = \frac{\hbar a}{2\pi k_B c}

where a is proper acceleration.

Quantum Fields Near Horizons

  1. Near-Horizon Modes In Kruskal coordinates:
uω∼e−iωU/4πωu_\omega \sim e^{-i\omega U}/\sqrt{4\pi\omega} vω∼e−iωV/4πωv_\omega \sim e^{-i\omega V}/\sqrt{4\pi\omega}
  1. Gravitational Redshift Frequency shift:
ω∞=ωlocal−g00\omega_\infty = \omega_{\text{local}}\sqrt{-g_{00}}

Advanced Topics

  1. Stress-Energy Renormalization Point-splitting regularization:
⟨Tμν⟩ren=lim⁡x′→xDμν(x,x′)G(1)(x,x′)+local terms\langle T_{\mu\nu}\rangle_{\text{ren}} = \lim_{x'\to x}D_{\mu\nu}(x,x')G^{(1)}(x,x') + \text{local terms}
  1. Trace Anomaly Quantum violation of classical conformal invariance:
⟨Tμμ⟩=12880π2(RμνρσRμνρσ−RμνRμν+□R)\langle T^\mu_\mu\rangle = \frac{1}{2880\pi^2}(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} - R_{\mu\nu}R^{\mu\nu} + \square R)
  1. Back-reaction Semi-classical Einstein equations:
Gμν=8πG⟨Tμν⟩renG_{\mu\nu} = 8\pi G\langle T_{\mu\nu}\rangle_{\text{ren}}

Physical Applications

  1. Black Hole Evaporation Mass loss rate:
dMdt=−ℏ15360πGM2\frac{dM}{dt} = -\frac{\hbar}{15360\pi GM^2}

Leading to finite lifetime:

τ∼G2M3ℏ>c4\tau \sim \frac{G^2M^3}{\hbar > c^4}
  1. Cosmological Particle Creation In expanding universe:
nk∼e−πωk/Hn_k \sim e^{-\pi\omega_k/H}

where H is Hubble parameter.

  1. Casimir Effect Energy density between plates:
ρ=−π2ℏc720d4\rho = -\frac{\pi^2\hbar c}{720d^4}

Modified by curvature.

Conceptual Challenges

  1. Trans-Planckian Problem Near horizon modes have arbitrarily high frequency:
ωlocal∼ω∞eκt\omega_{\text{local}} \sim \omega_\infty e^{\kappa t}

Requiring quantum gravity input.

  1. Information Paradox Pure states evolve to mixed states:
ρfinal=Trinside∣ψ⟩⟨ψ∣\rho_{\text{final}} = \text{Tr}_{\text{inside}}|\psi\rangle\langle\psi|

Violating quantum mechanics.

  1. Firewall Paradox Tension between:

    • Equivalence principle

    • Unitarity

    • Local quantum field theory

Exercises for Understanding

  1. Mode Analysis Find normalized modes for scalar field in FRW universe:
ds2=−dt2+a2(t)(dx2+dy2+dz2)ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)
  1. Particle Creation Calculate Bogoliubov coefficients for sudden transition:
a(t)={1t<02t>0a(t) = \begin{cases} 1 & t < 0 \\ 2 & t > 0 \end{cases}
  1. Hawking Radiation Derive temperature for Reissner-Nordström black hole:
TH=κ2π=r+−r−4πr+2T_H = \frac{\kappa}{2\pi} = \frac{r_+ - r_-}{4\pi r_+^2}

Looking Forward

This concludes our study of multivariable calculus, but we’ve seen how these mathematical tools extend to quantum field theory in curved spacetime. Next, we’ll begin exploring:

  • Differential geometry

  • Lie groups and algebras

  • Fiber bundles

These topics will provide the mathematical framework needed for a full theory of quantum gravity.

Remember: Quantum fields in curved spacetime represent our best current understanding of how quantum matter interacts with gravity. The fascinating phenomena we’ve studied here provide crucial hints about the nature of quantum gravity itself.

End of Lesson 4: Multivariable Calculus II