Linearized Gravity

The weak-field perturbative expansion of general relativity around flat Minkowski spacetime — the foundation upon which the entire theory of gravitational waves is built.

1. Introduction: The Weak-Field Limit

The full Einstein field equations are a set of ten coupled, nonlinear partial differential equations for the metric tensor. Exact solutions exist only for highly symmetric configurations (Schwarzschild, Kerr, FLRW). For most physical scenarios — especially gravitational wave propagation far from sources — we work in the weak-field limit, where spacetime is nearly flat.

The central idea is to decompose the full metric into a flat background plus a small perturbation:

$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \qquad |h_{\mu\nu}| \ll 1$

where $\eta_{\mu\nu} = \mathrm{diag}(-1,+1,+1,+1)$ is the Minkowski metric and $h_{\mu\nu}$ is the perturbation.

This perturbative approach is valid whenever the gravitational field is weak: far from compact objects, in the solar system (where $|h| \sim GM/rc^2 \sim 10^{-6}$ at Earth's surface), and crucially for gravitational waves passing through our detectors (where $|h| \sim 10^{-21}$).

The strategy of linearized gravity is straightforward:

Expand all geometric quantities (Christoffel symbols, Riemann tensor, etc.) to first order in $h_{\mu\nu}$.
Discard all terms quadratic or higher in $h$.
Obtain a set of linear partial differential equations for $h_{\mu\nu}$.
Exploit gauge freedom to simplify these equations into a wave equation.

Key Assumptions

●Weak field: $|h_{\mu\nu}| \ll 1$ everywhere in the region of interest.
●Asymptotic flatness: $h_{\mu\nu} \to 0$ as $r \to \infty$.
●Coordinate choice: We work in nearly Cartesian coordinates $(t, x, y, z)$.
●Index raising/lowering: In linearized theory, indices are raised and lowered with $\eta_{\mu\nu}$ (not $g_{\mu\nu}$), since corrections would be second order.

2. Derivation: Linearized Einstein Equations

We now carry out the full linearization of Einstein's equations step by step. Starting from $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, we compute each geometric object to first order in $h$.

Step 1: Inverse Metric

The inverse metric must satisfy $g^{\mu\alpha}g_{\alpha\nu} = \delta^\mu_\nu$. Writing $g^{\mu\nu} = \eta^{\mu\nu} + \delta g^{\mu\nu}$ and expanding to first order:

$(\eta^{\mu\alpha} + \delta g^{\mu\alpha})(\eta_{\alpha\nu} + h_{\alpha\nu}) = \delta^\mu_\nu$

$\delta^\mu_\nu + \eta^{\mu\alpha}h_{\alpha\nu} + \delta g^{\mu\alpha}\eta_{\alpha\nu} + \mathcal{O}(h^2) = \delta^\mu_\nu$

$\Rightarrow \quad \delta g^{\mu\nu} = -\eta^{\mu\alpha}\eta^{\nu\beta}h_{\alpha\beta} \equiv -h^{\mu\nu}$

$\boxed{g^{\mu\nu} = \eta^{\mu\nu} - h^{\mu\nu} + \mathcal{O}(h^2)}$

Step 2: Christoffel Symbols to First Order

The Christoffel symbols are defined as:

$\Gamma^\lambda_{\mu\nu} = \frac{1}{2}g^{\lambda\sigma}\left(\partial_\mu g_{\sigma\nu} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu}\right)$

Since $\partial_\mu \eta_{\alpha\beta} = 0$ (flat background in Cartesian coordinates), we have $\partial_\mu g_{\alpha\beta} = \partial_\mu h_{\alpha\beta}$. Using $g^{\lambda\sigma} = \eta^{\lambda\sigma} - h^{\lambda\sigma}$ and keeping only first-order terms:

$\Gamma^\lambda_{\mu\nu} = \frac{1}{2}\eta^{\lambda\sigma}\left(\partial_\mu h_{\sigma\nu} + \partial_\nu h_{\sigma\mu} - \partial_\sigma h_{\mu\nu}\right) + \mathcal{O}(h^2)$

$\boxed{\Gamma^\lambda_{\mu\nu} = \frac{1}{2}\left(\partial_\mu h^\lambda{}_\nu + \partial_\nu h^\lambda{}_\mu - \partial^\lambda h_{\mu\nu}\right)}$

Note: the $-h^{\lambda\sigma}$ piece of $g^{\lambda\sigma}$ multiplied by $\partial h$ gives an $\mathcal{O}(h^2)$ term, which we discard.

Step 3: Riemann Tensor to First Order

The Riemann tensor is:

$R^\lambda{}_{\mu\nu\sigma} = \partial_\nu\Gamma^\lambda_{\mu\sigma} - \partial_\sigma\Gamma^\lambda_{\mu\nu} + \Gamma^\lambda_{\alpha\nu}\Gamma^\alpha_{\mu\sigma} - \Gamma^\lambda_{\alpha\sigma}\Gamma^\alpha_{\mu\nu}$

Since $\Gamma \sim \mathcal{O}(h)$, the quadratic $\Gamma\Gamma$ terms are $\mathcal{O}(h^2)$ and we drop them. Thus:

$R^{(1)\lambda}{}_{\mu\nu\sigma} = \partial_\nu\Gamma^\lambda_{\mu\sigma} - \partial_\sigma\Gamma^\lambda_{\mu\nu}$

Substituting our linearized Christoffel symbols and lowering the first index with $\eta_{\lambda\rho}$:

$\boxed{R^{(1)}_{\lambda\mu\nu\sigma} = \frac{1}{2}\left(\partial_\nu\partial_\mu h_{\lambda\sigma} + \partial_\sigma\partial_\lambda h_{\mu\nu} - \partial_\sigma\partial_\mu h_{\lambda\nu} - \partial_\nu\partial_\lambda h_{\mu\sigma}\right)}$

This is manifestly antisymmetric in $(\lambda,\mu)$ and $(\nu,\sigma)$, as required.

Step 4: Ricci Tensor

Contracting $\lambda$ with $\nu$:

$R^{(1)}_{\mu\sigma} = \eta^{\lambda\nu}R^{(1)}_{\lambda\mu\nu\sigma}$

$= \frac{1}{2}\left(\partial^\nu\partial_\mu h_{\nu\sigma} + \partial_\sigma\partial^\nu h_{\mu\nu} - \partial_\sigma\partial_\mu h - \Box h_{\mu\sigma}\right)$

where $h \equiv \eta^{\mu\nu}h_{\mu\nu} = h^\mu{}_\mu$ is the trace and$\Box \equiv \eta^{\mu\nu}\partial_\mu\partial_\nu = -\frac{1}{c^2}\frac{\partial^2}{\partial t^2} + \nabla^2$ is the d'Alembertian (wave operator). Therefore:

$\boxed{R^{(1)}_{\mu\nu} = \frac{1}{2}\left(\partial^\alpha\partial_\mu h_{\alpha\nu} + \partial^\alpha\partial_\nu h_{\alpha\mu} - \partial_\mu\partial_\nu h - \Box h_{\mu\nu}\right)}$

Step 5: Ricci Scalar

$R^{(1)} = \eta^{\mu\nu}R^{(1)}_{\mu\nu} = \partial^\mu\partial^\nu h_{\mu\nu} - \Box h$

Step 6: Linearized Einstein Tensor

The Einstein tensor is $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$. To first order,$g_{\mu\nu}R = \eta_{\mu\nu}R^{(1)} + \mathcal{O}(h^2)$, so:

$G^{(1)}_{\mu\nu} = R^{(1)}_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}R^{(1)}$

$= \frac{1}{2}\Big(\partial^\alpha\partial_\mu h_{\alpha\nu} + \partial^\alpha\partial_\nu h_{\alpha\mu} - \partial_\mu\partial_\nu h - \Box h_{\mu\nu} - \eta_{\mu\nu}\partial^\alpha\partial^\beta h_{\alpha\beta} + \eta_{\mu\nu}\Box h\Big)$

The linearized Einstein equations are then:

$\boxed{G^{(1)}_{\mu\nu} = \frac{8\pi G}{c^4}\, T_{\mu\nu}}$

The linearized Einstein field equations: a set of linear PDEs for $h_{\mu\nu}$.

Remark: The stress-energy tensor $T_{\mu\nu}$ on the right-hand side is itself treated as a first-order quantity. The equations are consistent at linear order: $\nabla^\mu T_{\mu\nu} = 0$ is enforced by the Bianchi identity $\nabla^\mu G_{\mu\nu} = 0$, which at linear order becomes$\partial^\mu G^{(1)}_{\mu\nu} = 0$. One can verify this directly from the expression above.

3. Trace-Reversed Perturbation and Wave Equation

The linearized Einstein equations in their raw form are unwieldy. A crucial simplification comes from introducing the trace-reversed perturbation.

Defining the Trace-Reversed Metric

$\bar{h}_{\mu\nu} \equiv h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h$

where $h = \eta^{\alpha\beta}h_{\alpha\beta}$ is the trace. The name “trace-reversed” comes from the property:

$\bar{h} \equiv \eta^{\mu\nu}\bar{h}_{\mu\nu} = h - \frac{1}{2}(4)h = h - 2h = -h$

The inverse relation is $h_{\mu\nu} = \bar{h}_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}\bar{h}$. Now we rewrite the linearized Einstein equations in terms of $\bar{h}_{\mu\nu}$. After substituting and simplifying (which involves considerable algebra), we obtain:

$\boxed{-\Box\bar{h}_{\mu\nu} - \eta_{\mu\nu}\partial^\alpha\partial^\beta\bar{h}_{\alpha\beta} + \partial^\alpha\partial_\mu\bar{h}_{\alpha\nu} + \partial^\alpha\partial_\nu\bar{h}_{\alpha\mu} = \frac{16\pi G}{c^4}\,T_{\mu\nu}}$

This is the linearized Einstein equation rewritten in terms of the trace-reversed perturbation.

To verify this, one substitutes $h_{\mu\nu} = \bar{h}_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}\bar{h}$ into the expression for $G^{(1)}_{\mu\nu}$. The key identities used are:

●$\Box h_{\mu\nu} = \Box\bar{h}_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}\Box\bar{h}$
●$\partial^\alpha\partial_\mu h_{\alpha\nu} = \partial^\alpha\partial_\mu\bar{h}_{\alpha\nu} - \frac{1}{2}\partial_\mu\partial_\nu\bar{h}$
●$\partial_\mu\partial_\nu h = -\partial_\mu\partial_\nu\bar{h}$
●$\partial^\alpha\partial^\beta h_{\alpha\beta} = \partial^\alpha\partial^\beta\bar{h}_{\alpha\beta} - \frac{1}{2}\Box\bar{h}$

Imposing the Lorenz Gauge

The three non-wave terms in the equation above can be eliminated by a gauge choice. We impose the Lorenz gauge condition (also called the de Donder gauge or harmonic gauge):

$\partial^\mu \bar{h}_{\mu\nu} = 0$

Four conditions (one for each value of $\nu$), analogous to the Lorenz gauge $\partial^\mu A_\mu = 0$ in electromagnetism.

With this gauge condition, $\partial^\alpha\bar{h}_{\alpha\nu} = 0$ and$\partial^\alpha\partial^\beta\bar{h}_{\alpha\beta} = 0$, so all the non-wave terms vanish identically. The linearized Einstein equations reduce to the beautifully simple form:

$\boxed{\Box\bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4}\,T_{\mu\nu}}$

A wave equation with source! In vacuum ($T_{\mu\nu} = 0$), this becomes $\Box\bar{h}_{\mu\nu} = 0$ — the equation for freely propagating gravitational waves at the speed of light.

Analogy with electromagnetism: This is the gravitational analog of$\Box A^\mu = -\mu_0 J^\mu$ in the Lorenz gauge $\partial_\mu A^\mu = 0$. The trace-reversed perturbation $\bar{h}_{\mu\nu}$ plays the role of the vector potential $A^\mu$, and the stress-energy tensor $T_{\mu\nu}$ plays the role of the current density $J^\mu$. The key difference: gravity involves a symmetric rank-2 tensor (spin-2) rather than a 4-vector (spin-1).

4. Gauge Freedom and Physical Degrees of Freedom

Just as in electromagnetism, the linearized theory possesses gauge freedom: different perturbations$h_{\mu\nu}$ can describe the same physical spacetime. Understanding this gauge freedom is essential for identifying the true physical content of gravitational waves.

Infinitesimal Coordinate Transformations

Consider an infinitesimal coordinate transformation:

$x^\mu \to x'^\mu = x^\mu + \xi^\mu(x), \qquad |\xi^\mu| \sim \mathcal{O}(h)$

Under this transformation, the metric transforms as:

$g'_{\mu\nu}(x') = \frac{\partial x^\alpha}{\partial x'^\mu}\frac{\partial x^\beta}{\partial x'^\nu}g_{\alpha\beta}(x)$

$= (\delta^\alpha_\mu - \partial_\mu\xi^\alpha)(\delta^\beta_\nu - \partial_\nu\xi^\beta)(\eta_{\alpha\beta} + h_{\alpha\beta})$

$= \eta_{\mu\nu} + h_{\mu\nu} - \partial_\mu\xi_\nu - \partial_\nu\xi_\mu + \mathcal{O}(h^2, \xi^2)$

Since the new metric in the new coordinates still has the form $\eta_{\mu\nu} + h'_{\mu\nu}$, we identify the gauge transformation of the perturbation:

$\boxed{h_{\mu\nu} \to h'_{\mu\nu} = h_{\mu\nu} - \partial_\mu\xi_\nu - \partial_\nu\xi_\mu}$

Compare with the EM gauge transformation: $A_\mu \to A_\mu - \partial_\mu\Lambda$.

For the trace-reversed perturbation, the transformation becomes:

$\bar{h}'_{\mu\nu} = \bar{h}_{\mu\nu} - \partial_\mu\xi_\nu - \partial_\nu\xi_\mu + \eta_{\mu\nu}\partial_\alpha\xi^\alpha$

Existence of the Lorenz Gauge

We can always find a gauge transformation that imposes the Lorenz condition. If we start with some $\bar{h}_{\mu\nu}$ that does not satisfy $\partial^\mu\bar{h}_{\mu\nu} = 0$, we need to find $\xi^\mu$ such that after the transformation:

$\partial^\mu\bar{h}'_{\mu\nu} = \partial^\mu\bar{h}_{\mu\nu} - \Box\xi_\nu = 0$

$\Rightarrow \quad \Box\xi_\nu = \partial^\mu\bar{h}_{\mu\nu}$

This is a wave equation for $\xi_\nu$ with a known source, which always has a solution. Thus the Lorenz gauge can always be reached.

Residual Gauge Freedom

The Lorenz gauge does not completely fix the gauge. Even after imposing $\partial^\mu\bar{h}_{\mu\nu} = 0$, we can still make further gauge transformations with any $\xi^\mu$ satisfying:

$\Box\xi^\mu = 0$

Any harmonic vector field $\xi^\mu$ preserves the Lorenz condition.

Counting Physical Degrees of Freedom

Symmetric tensor $h_{\mu\nu}$ in 4D	$\frac{4 \times 5}{2} = 10$ components
Lorenz gauge conditions $\partial^\mu\bar{h}_{\mu\nu} = 0$	−4 constraints
Residual gauge freedom ($\Box\xi^\mu = 0$)	−4 more
Physical degrees of freedom	2

The two physical polarization states correspond to the plus (+) and cross (×) polarization modes of gravitational waves. This is consistent with the fact that gravitational waves are described by a massless spin-2 field, which in four dimensions has exactly two helicity states ($\pm 2$).

Comparison with electromagnetism: The photon (spin-1, massless) has a 4-vector potential $A_\mu$ with 4 components. Lorenz gauge ($\partial^\mu A_\mu = 0$) removes 1, and residual gauge ($\Box\Lambda = 0$) removes 1 more, leaving $4 - 1 - 1 = 2$ physical polarizations. The pattern is identical: a massless spin-$s$ field in 4D always has exactly 2 physical polarizations (helicities $\pm s$).

5. Green's Function Solution

Having reduced the linearized Einstein equations to a wave equation in the Lorenz gauge, we can solve it using the retarded Green's function — exactly as in classical electrodynamics.

The Retarded Green's Function

The Green's function for the d'Alembertian satisfies:

$\Box_x\, G(x - x') = \delta^{(4)}(x - x')$

The retarded Green's function, which respects causality (effects propagate forward in time), is:

$\boxed{G_{\mathrm{ret}}(x - x') = -\frac{1}{4\pi|\mathbf{x} - \mathbf{x}'|}\,\delta\!\left(t' - t_{\mathrm{ret}}\right)}$

where the retarded time is $t_{\mathrm{ret}} = t - \frac{|\mathbf{x} - \mathbf{x}'|}{c}$.

The Retarded Solution

The general solution to $\Box\bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4}T_{\mu\nu}$ with outgoing-wave boundary conditions is obtained by convolving with the retarded Green's function:

$\bar{h}_{\mu\nu}(t, \mathbf{x}) = -\frac{16\pi G}{c^4}\int G_{\mathrm{ret}}(x - x')\, T_{\mu\nu}(x')\, d^4x'$

$= \frac{4G}{c^4}\int \frac{T_{\mu\nu}(t_{\mathrm{ret}}, \mathbf{x}')}{|\mathbf{x} - \mathbf{x}'|}\, d^3x'$

$\boxed{\bar{h}_{\mu\nu}(t, \mathbf{x}) = \frac{4G}{c^4}\int \frac{T_{\mu\nu}\!\left(t - \frac{|\mathbf{x} - \mathbf{x}'|}{c},\, \mathbf{x}'\right)}{|\mathbf{x} - \mathbf{x}'|}\, d^3x'}$

The gravitational retarded potential — the direct analog of the Liénard-Wiechert potential in electromagnetism.

This formula encapsulates the key physics of gravitational wave generation:

●Source: The perturbation is generated by the stress-energy tensor $T_{\mu\nu}$ of matter.
●Retardation: The source is evaluated at the retarded time $t_{\mathrm{ret}} = t - |\mathbf{x}-\mathbf{x}'|/c$, reflecting finite propagation speed.
●1/r falloff: The amplitude falls off as $1/|\mathbf{x}-\mathbf{x}'|$, characteristic of radiation in 3+1 dimensions.
●Coupling constant: The prefactor $4G/c^4 \approx 3.3 \times 10^{-44}\;\mathrm{s^2\,kg^{-1}\,m^{-1}}$ is extraordinarily small, explaining why gravitational waves are so weak.

Far-Field (Radiation Zone) Limit

When the observer is far from the source ($r = |\mathbf{x}| \gg d$, where $d$ is the source size), we can approximate $|\mathbf{x} - \mathbf{x}'| \approx r - \hat{\mathbf{n}}\cdot\mathbf{x}'$(where $\hat{\mathbf{n}} = \mathbf{x}/r$). In the denominator, we keep only the leading term $1/r$:

$\bar{h}_{\mu\nu}(t, \mathbf{x}) \approx \frac{4G}{c^4 r}\int T_{\mu\nu}\!\left(t - \frac{r}{c} + \frac{\hat{\mathbf{n}}\cdot\mathbf{x}'}{c},\, \mathbf{x}'\right) d^3x'$

For non-relativistic sources (where the internal velocities $v \ll c$ and the source size $d \ll \lambda_{\mathrm{GW}}$), the retardation across the source is negligible. This leads to the multipole expansion and ultimately the quadrupole formula, covered in Chapter 4.

6. Applications of Linearized Gravity

6.1 Recovery of the Newtonian Limit

For a static, non-relativistic source with energy density $\rho$ and negligible pressure and momentum, $T_{00} = \rho c^2$ and all other components are small. The wave equation reduces to Poisson's equation:

$\nabla^2 \bar{h}_{00} = -\frac{16\pi G}{c^2}\rho$

$\Rightarrow \quad \bar{h}_{00} = \frac{4\Phi}{c^2}, \quad \text{where } \nabla^2\Phi = 4\pi G\rho$

Converting back to the original perturbation using $h_{\mu\nu} = \bar{h}_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}\bar{h}$:

$\boxed{h_{00} = -\frac{2\Phi}{c^2}, \qquad h_{ij} = -\frac{2\Phi}{c^2}\,\delta_{ij}}$

This gives the weak-field metric: $ds^2 = -(1 + 2\Phi/c^2)c^2 dt^2 + (1 - 2\Phi/c^2)(dx^2 + dy^2 + dz^2)$.

This is precisely the Newtonian limit of general relativity, confirming that linearized theory correctly reproduces Newtonian gravity as its static limit.

6.2 Gravitoelectromagnetism

When the source has a mass current (momentum density) $T_{0i} = \rho v_i$, the off-diagonal components $\bar{h}_{0i}$ become non-zero. Defining gravitoelectric and gravitomagnetic fields by analogy with electromagnetism:

$\mathbf{E}_g = -\nabla\Phi - \frac{1}{c}\frac{\partial\mathbf{A}_g}{\partial t}, \qquad \mathbf{B}_g = \nabla \times \mathbf{A}_g$

where $\Phi = -c^2 h_{00}/2$ and $A_{g,i} = -c^2 h_{0i}/2$.

These fields obey equations analogous to Maxwell's equations (with factors of 4 due to the tensor nature of gravity). The gravitomagnetic field $\mathbf{B}_g$ is responsible for the Lense-Thirring effect (frame dragging), which has been confirmed by Gravity Probe B and by observations of orbiting pulsars.

6.3 Frame Dragging

For a slowly rotating body with angular momentum $\mathbf{J}$, the gravitomagnetic potential at distance $r$ is:

$\mathbf{A}_g = \frac{G}{c}\frac{\mathbf{J} \times \hat{\mathbf{r}}}{r^2}$

$h_{0i} = -\frac{2}{c^2}A_{g,i} = -\frac{2G}{c^3}\frac{(\mathbf{J}\times\hat{\mathbf{r}})_i}{r^2}$

This causes precession of gyroscopes and orbital planes at a rate:

$\mathbf{\Omega}_{\mathrm{LT}} = \frac{G}{c^2 r^3}\left[3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}} - \mathbf{J}\right]$

Gravity Probe B measured this precession rate for gyroscopes in Earth orbit, confirming the prediction to within 19% (limited by systematic effects in the experiment). The LAGEOS satellite measurements have confirmed frame dragging to approximately 10%.

7. Historical Context

Einstein (1916): The Original Prediction

Einstein published his linearized analysis of gravitational waves in June 1916, just months after completing general relativity. He showed that the linearized vacuum equations admit wave solutions propagating at the speed of light. However, his initial paper contained an error: he found three polarization modes instead of two (he had not fully exploited gauge freedom). He corrected this in a subsequent paper in 1918, where he also derived the quadrupole formula for gravitational wave emission.

The Reality Debate (1936–1957)

Einstein himself later doubted the physical reality of gravitational waves! In a 1936 paper with Rosen, he argued that exact gravitational wave solutions are impossible (the paper was rejected by Physical Review — one of the rare rejections Einstein received). The referee (later identified as Howard Robertson) found the error: Einstein and Rosen had confused a coordinate singularity for a physical one. The debate was largely settled at the 1957 Chapel Hill conference, where Felix Pirani showed that gravitational waves produce measurable geodesic deviation, and Richard Feynman gave his famous “sticky bead argument” demonstrating that gravitational waves carry energy.

Comparison with Electromagnetic Radiation Theory

The mathematical structure of linearized gravity closely parallels that of electromagnetism:

Feature	Electromagnetism	Linearized Gravity
Field	$A_\mu$ (4-vector)	$\bar{h}_{\mu\nu}$ (symmetric tensor)
Source	$J^\mu$ (4-current)	$T_{\mu\nu}$ (stress-energy)
Gauge	$\partial^\mu A_\mu = 0$	$\partial^\mu\bar{h}_{\mu\nu} = 0$
Wave equation	$\Box A^\mu = -\mu_0 J^\mu$	$\Box\bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4}T_{\mu\nu}$
Spin	1 (vector boson)	2 (tensor boson)
Polarizations	2	2
Leading multipole	Dipole	Quadrupole

The absence of gravitational dipole radiation is a consequence of conservation of momentum: the mass dipole moment $\sum m_a \mathbf{x}_a$ satisfies $\ddot{D}_i = 0$ for an isolated system. This is fundamentally different from EM, where charges can have a net oscillating dipole moment.

8. Python Simulation: Binary Metric Perturbation

This simulation computes the linearized metric perturbation $h_{00}$ from a rotating equal-mass binary system, using the retarded Green's function solution derived above. We visualize:

●The full $h_{00}$ perturbation and its decomposition into monopole + wave parts.
●The $1/r$ falloff of the wave perturbation (verified by checking that $r \cdot h_{\mathrm{wave}}$ oscillates around a constant).
●The angular structure of the radiation pattern.
●Retardation effects: the phase shift between signals at different radii due to finite propagation speed.

Linearized Metric Perturbation from a Rotating Binary

Python

Computes h_00 using the retarded Green's function for two equal masses on a circular orbit. Demonstrates 1/r falloff, retardation, and angular radiation pattern.

linearized_binary_perturbation.py99 lines

import numpy as np

# ========================================================
# Linearized metric perturbation from a rotating binary
# ========================================================
# We compute the trace-reversed perturbation h_bar_{00}
# from two equal masses on a circular orbit using the
# retarded Green's function solution:
#   h_bar_{mu nu}(t,x) = (4G/c^4) int T_{mu nu}(t_ret,x') / |x-x'| d^3x'
# For the 00-component with non-relativistic sources,
# h_bar_{00} ~ (4G/c^2) int rho(t_ret,x') / |x-x'| d^3x'
# which reduces to h_00 = -2 Phi/c^2 plus wave corrections.

# Physical parameters (scaled units: G=1, c=1)
M = 1.0           # total mass
a = 10.0           # orbital separation
omega = np.sqrt(M / a**3)  # Kepler frequency
m1 = m2 = M / 2.0

# Observer grid in the x-z plane
Nr = 200
Ntheta = 200
r_min, r_max = 15.0, 120.0
r_vals = np.linspace(r_min, r_max, Nr)
theta_vals = np.linspace(0, 2 * np.pi, Ntheta)
R, THETA = np.meshgrid(r_vals, theta_vals)
X_obs = R * np.cos(THETA)
Z_obs = R * np.sin(THETA)

# Snapshot time
t_obs = 0.0

# Compute the Newtonian potential plus leading retardation correction
# Source positions at retarded time for each mass
h00 = np.zeros_like(R)

for i_r in range(Nr):
    for i_th in range(Ntheta):
        xo = X_obs[i_th, i_r]
        zo = Z_obs[i_th, i_r]

# Distance from origin to observer point
        r_obs = np.sqrt(xo**2 + zo**2)

# Retarded time (approximate: t_ret ~ t - r_obs/c, c=1)
        t_ret = t_obs - r_obs

# Source 1 position at retarded time (in x-y orbital plane, y=0 slice)
        x1 = (a / 2) * np.cos(omega * t_ret)
        y1 = (a / 2) * np.sin(omega * t_ret)
        # Source 2 opposite
        x2 = -x1
        y2 = -y1

# 3D distances (observer in x-z plane, sources in x-y plane)
        d1 = np.sqrt((xo - x1)**2 + y1**2 + zo**2)
        d2 = np.sqrt((xo - x2)**2 + y2**2 + zo**2)

# h_00 = -2 Phi / c^2 = 2G(m1/d1 + m2/d2) / c^2 (G=c=1)
        h00[i_th, i_r] = 2.0 * (m1 / d1 + m2 / d2)

# Subtract the static monopole (1/r) part to isolate wave perturbation
h00_monopole = 2.0 * M / R
h00_wave = h00 - h00_monopole

# ---- Plot 1: Full h_00 perturbation along theta=0 (equatorial) ----
idx_eq = 0  # theta = 0
print("# r  h00_full  h00_monopole  h00_wave")
step = max(1, Nr // 60)
for i in range(0, Nr, step):
    print(f"{r_vals[i]:.4f}  {h00[idx_eq, i]:.8f}  {h00_monopole[idx_eq, i]:.8f}  {h00_wave[idx_eq, i]:.8f}")

print()
print("# --- 1/r falloff verification ---")
print("# r  r*h00_wave  (should oscillate around constant if ~ 1/r)")
for i in range(0, Nr, step):
    val = r_vals[i] * h00_wave[idx_eq, i]
    print(f"{r_vals[i]:.4f}  {val:.8f}")

print()
print("# --- Angular profile at fixed r ---")
r_idx_fixed = Nr // 2
r_fixed = r_vals[r_idx_fixed]
print(f"# theta  h00_wave  at r={r_fixed:.1f}")
step_th = max(1, Ntheta // 60)
for j in range(0, Ntheta, step_th):
    print(f"{theta_vals[j]:.4f}  {h00_wave[j, r_idx_fixed]:.8f}")

print()
print("# --- Retardation phase shift ---")
print("# Comparing h00_wave at two radii vs theta")
r_idx1 = Nr // 4
r_idx2 = 3 * Nr // 4
r1_val = r_vals[r_idx1]
r2_val = r_vals[r_idx2]
print(f"# theta  h00_wave(r={r1_val:.1f})  h00_wave(r={r2_val:.1f})")
for j in range(0, Ntheta, step_th):
    print(f"{theta_vals[j]:.4f}  {h00_wave[j, r_idx1]:.8f}  {h00_wave[j, r_idx2]:.8f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Summary of Key Results

Metric Decomposition

$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \quad |h_{\mu\nu}| \ll 1$

Linearized Christoffel Symbols

$\Gamma^\lambda_{\mu\nu} = \frac{1}{2}(\partial_\mu h^\lambda{}_\nu + \partial_\nu h^\lambda{}_\mu - \partial^\lambda h_{\mu\nu})$

Trace-Reversed Perturbation

$\bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h$

Wave Equation (Lorenz Gauge)

$\Box\bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4}T_{\mu\nu}, \quad \partial^\mu\bar{h}_{\mu\nu} = 0$

Gauge Transformation

$h_{\mu\nu} \to h_{\mu\nu} - \partial_\mu\xi_\nu - \partial_\nu\xi_\mu$

Retarded Solution

$\bar{h}_{\mu\nu}(t,\mathbf{x}) = \frac{4G}{c^4}\int\frac{T_{\mu\nu}(t_{\mathrm{ret}},\mathbf{x}')}{|\mathbf{x}-\mathbf{x}'|}d^3x'$

Physical Degrees of Freedom

10 (symmetric tensor) − 4 (Lorenz gauge) − 4 (residual gauge) = 2 polarizations (plus and cross)

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