Covariant Radiative Transfer

Covariant radiative transfer follows massless particles on the future null shell while allowing emission, absorption, and scattering to change their phase-space distribution. The central invariant is $I_\nu/\nu^3$: specific intensity itself changes under gravitational redshift or an observer boost, but this ratio is a scalar along a collisionless ray. That fact separates spacetime propagation from local microphysics and supplies immediate checks on every ray-tracing or transport calculation.

Required background. Hilbert Stress–Energy from Metric Variation supplies radiation stress and exchange; Differential Equations supplies transport along characteristics.

Helpful background. The Einstein Equation supplies curved-spacetime propagation; Relativistic Matter and Compact Objects gives the transport context; and Computational Relativity supplies geodesic and evolution methods.

Radiation on the null mass shell

A photon four-momentum satisfies

$$p^ap_a=0, \qquad p^a\text{ future directed}.$$

An observer with four-velocity $u^a$ measures photon energy $\epsilon$ and ordinary frequency $\nu$ according to

$$\epsilon=-p_a u^a\gt0, \qquad \epsilon=h\nu=2\pi\nu.$$

The last equality uses the site’s $\hbar=1$ convention; energy equals angular frequency, not ordinary frequency. Factors of $h$ cancel from frequency ratios but matter when normalizing a spectrum. The occupation distribution $f(x,p)$ is a scalar on the null shell. Between interactions it obeys Liouville transport,

$$p^a\frac{\partial f}{\partial x^a} -\Gamma^i{}_{ab}p^ap^b\frac{\partial f}{\partial p^i}=0,$$

so $f$ is constant along an affinely parametrized null geodesic. With a fixed polarization convention, specific intensity is proportional to $\nu^3f$; hence

$$\mathcal I\equiv\frac{I_\nu}{\nu^3}$$

is invariant under changes of local observer and constant along a collisionless ray. The phase-space argument and transfer invariants are developed in Younsi, Wu, and Fuerst 2012, § 2, Eqs. (5)–(9).

This normalization has a direct dimensional check. Specific intensity is energy per area, time, solid angle, and frequency interval; the phase-space occupation is dimensionless after the conventional constants are included. A different convention may place factors of $2$, $h$, or polarization multiplicity in the proportionality, but it cannot change the $\nu^3$ transformation law.

Emission, absorption, and scattering

Local microphysics is most simply specified in an orthonormal frame comoving with matter. In a stationary flat-space medium, with frequency fixed along a ray and path length $\ell$, pure emission and absorption obey

$$\frac{dI_\nu}{d\ell}=j_\nu-\alpha_\nu I_\nu,$$

where $j_\nu$ is emissivity and $\alpha_\nu$ absorption coefficient. In a moving medium or curved spacetime, the comoving frequency changes along the ray; the preceding derivative cannot then be used as the full propagation law for $I_\nu$. Instead evolve the invariant intensity:

$$\frac{d\mathcal I}{d\ell}=\frac{j_\nu}{\nu^3}-\alpha_\nu\mathcal I, \qquad d\ell=-u_ap^a\,d\lambda,$$

where $dx^a/d\lambda=p^a$ is the future-directed affine tangent and $d\ell$ is the infinitesimal path length measured by the local material observer. All collision coefficients and $\nu$ on the right are evaluated in that same frame. The invariants $j_\nu/\nu^2$ and $\nu\alpha_\nu$ make the equation independent of an auxiliary observer choice. See Younsi, Wu, and Fuerst 2012, § 2.1, Eqs. (13)–(18).

For constant coefficients in flat spacetime,

$$I_\nu(\ell)=I_\nu(0)e^{-\tau_\nu} +S_\nu\left(1-e^{-\tau_\nu}\right), \qquad \tau_\nu=\alpha_\nu\ell, \qquad S_\nu=\frac{j_\nu}{\alpha_\nu}.$$

For $\alpha_\nu\gt0$, this exact attenuation solution checks signs: with no emission intensity decays, while an optically thick homogeneous region approaches the source function. If $\alpha_\nu=0$, take the nonsingular limit $I_\nu(\ell)=I_\nu(0)+j_\nu\ell$ instead of dividing by zero. In thermal equilibrium detailed balance makes $S_\nu$ the Planck function in the material frame.

Scattering redistributes direction and possibly frequency. Its collision integral contains gain into and loss out of a momentum state. Coherent scattering preserves photon energy in the rest frame in which recoil is neglected; Compton scattering generally exchanges energy with electrons. Moving scatterers can therefore change photon frequency even when each event is treated as coherent in the scatterer’s frame. Collision terms must preserve the intended equilibrium and combined matter–radiation energy, not photon energy alone.

Redshift and observed intensity

Let the same ray be measured at emission and observation, with

$$g\equiv\frac{\nu_{\rm obs}}{\nu_{\rm em}} =\frac{(p_a u^a)_{\rm obs}}{(p_a u^a)_{\rm em}}.$$

In vacuum, invariance of $\mathcal I$ gives

$$I_{\nu,\rm obs}(\nu_{\rm obs}) =g^3 I_{\nu,\rm em}(\nu_{\rm em}).$$

After integrating over frequency, $d\nu_{\rm obs}=g d\nu_{\rm em}$ supplies a fourth factor, so bolometric intensity transforms as $I_{\rm obs}=g^4I_{\rm em}$ for the same ray bundle. Lens magnification is a separate mapping of solid angle and image area; it must not be folded into the invariant intensity law.

In a stationary region where a Killing field $t^a$ is timelike, $-p_at^a$ is conserved along the ray. Observers following that field have $u^a=t^a/\sqrt{-t^bt_b}$, so their measured frequency is inversely proportional to $\sqrt{-t^bt_b}$. For static observers outside a Schwarzschild mass $M$,

$$g=\sqrt{\frac{1-2GM/r_{\rm em}}{1-2GM/r_{\rm obs}}}, \qquad r_{\rm em},r_{\rm obs}\gt2GM.$$

A photon climbing outward is redshifted. In a time-dependent spacetime no conserved Killing energy need exist, but the local definition $h\nu=-p_au^a$ and Liouville theorem remain valid.

Radiation moments and limitations

Momentum moments of $f$ give a radiation stress tensor

$$R^{ab}=g_{\rm pol}\int p^ap^b f\,dP, \qquad dP=\frac{d^3p}{(2\pi)^3p^0}$$

in a local orthonormal frame. Here $g_{\rm pol}=2$ for unpolarized photons when $f$ is the occupation per polarization, and $p^0=|\boldsymbol p|$ on the future null shell. This convention gives $I_\nu=g_{\rm pol}h\nu^3f$ for equal occupations of the polarizations: inserting $p=h\nu$ and $d^3p=p^2dp\,d\Omega$ in the energy moment fixes the prefactor. Its divergence equals the negative four-force transferred to matter. Integrating the transfer equation therefore connects angle- and frequency-resolved radiation to energy–momentum exchange. Closing the moment hierarchy requires additional angular information, developed on Radiation Moments and Closure.

Geometric optics assumes wavelength is short compared with curvature and medium-variation scales. Coherent wave interference, polarization transport beyond a specified basis, refractive media, and quantum correlations can require extensions. Quantum occupation factors such as stimulated emission can already enter a kinetic collision operator; they do not by themselves invalidate geometric optics. Even within that limit, grey opacities can erase essential frequency dependence, and a comoving opacity used as though it were an observer-frame scalar produces incorrect redshift factors.

Exercises

A vacuum ray is redshifted by $g=1/2$. If its emitted specific intensity at the corresponding emitted frequency is $I_{\nu,\rm em}$, find the observed specific and bolometric intensity factors.

Solution

Invariant $I_\nu/\nu^3$ gives $I_{\nu,\rm obs}=g^3I_{\nu,\rm em}=I_{\nu,\rm em}/8$. Frequency integration contributes $d\nu_{\rm obs}=g d\nu_{\rm em}$, so $I_{\rm obs}=g^4I_{\rm em}=I_{\rm em}/16$. These factors describe intensity along the ray, not lensing magnification.

References

• Younsi, Ziri, Kinwah Wu, and Steven V. Fuerst. “General Relativistic Radiative Transfer: Formulation and Emission from Structured Tori around Black Holes.” Astronomy & Astrophysics 545, A13 (2012). DOI. Open preprint.