Linearized Einstein Equations: Weak-Field Limit

1. The Weak-Field Approximation

The core idea is to treat spacetime as being "almost flat." This means we can find a coordinate system where the metric $g_{\mu \nu }$ is just the flat Minkowski metric ${\eta }_{\mu \nu }$ plus a small perturbation, $h_{\mu \nu }$.

We formally express this as:
$g_{\mu \nu }(x)={\eta }_{\mu \nu }+h_{\mu \nu }(x)$
where $|h_{\mu \nu }|\ll 1$. We will work in coordinates where ${\eta }_{\mu \nu }=\text{diag}(-1,1,1,1)$.

This approximation, known as linearized gravity, means we will systematically neglect all terms of order $h^2$ or higher in our equations.

The Inverse Metric

Our first task is to find the inverse metric $g^{\mu \nu }$ consistent with this approximation. By definition, $g^{\mu \nu }{g}_{\nu \sigma }={\delta }_{\sigma }^{\mu }$. We can write $g^{\mu \nu }={\eta }^{\mu \nu }+k^{\mu \nu }$ for some small perturbation $k^{\mu \nu }$, and solve for $k^{\mu \nu }$ to first order in the perturbations. A quicker way is to note the analogy to the scalar binomial expansion $(1+ϵ{)}^{-1}\approx 1-ϵ$. The tensor equivalent is:
$g^{\mu \nu }\approx {\eta }^{\mu \nu }-h^{\mu \nu }$
where the indices on $h^{\mu \nu }$ are raised using the flat metric: $h^{\mu \nu }={\eta }^{\mu \alpha }{\eta }^{\nu \beta }{h}_{\alpha \beta }$.

In this approximation, we will consistently use ${\eta }_{\mu \nu }$ and ${\eta }^{\mu \nu }$ to lower and raise indices of any quantity that is already of order $h$. Including the $h_{\mu \nu }$ part of the full metric $g_{\mu \nu }$ for this task would produce terms of order $h^2$, which we neglect.

2. Linearizing Curvature

Now we apply this approximation to the machinery of curvature. The process is a "straightforward but tedious" calculation, as Prof. Scott Hughes aptly puts it in the MIT lecture. We will linearize the Christoffel symbols, then the Riemann tensor, and finally the Ricci tensor and scalar.

Christoffel Symbols:
The definition is ${\Gamma }_{\mu \nu }^{\rho }=\frac{1}{2}{g}^{\rho \lambda }({\partial }_{\mu }{g}_{\nu \lambda }+{\partial }_{\nu }{g}_{\lambda \mu }-{\partial }_{\lambda }{g}_{\mu \nu })$. In the linear approximation:

• $g^{\rho \lambda }\approx {\eta }^{\rho \lambda }$
• ${\partial }_{\alpha }{g}_{\beta \gamma }={\partial }_{\alpha }({\eta }_{\beta \gamma }+h_{\beta \gamma })={\partial }_{\alpha }{h}_{\beta \gamma }$ (since ${\eta }_{\beta \gamma }$ is constant).

This gives the linearized Christoffel symbol, which is first-order in $h$:
${\Gamma }_{\mu \nu }^{\rho }=\frac{1}{2}{\eta }^{\rho \lambda }({\partial }_{\mu }{h}_{\nu \lambda }+{\partial }_{\nu }{h}_{\lambda \mu }-{\partial }_{\lambda }{h}_{\mu \nu })$

Riemann and Ricci Tensors:
The Riemann tensor is $R_{\sigma \mu \nu }^{\rho }={\partial }_{\mu }{\Gamma }_{\nu \sigma }^{\rho }-{\partial }_{\nu }{\Gamma }_{\mu \sigma }^{\rho }+{\Gamma }_{\mu \sigma }^{\alpha }{\Gamma }_{\nu \alpha }^{\rho }-{\Gamma }_{\nu \sigma }^{\alpha }{\Gamma }_{\mu \alpha }^{\rho }$.
Since $\Gamma$ is already of order $h$, the $\Gamma \Gamma$ terms are of order $h^2$ and can be dropped immediately. This is a massive simplification! The entire curvature tensor comes only from the derivatives of the Christoffel symbols.

By substituting the linearized $\Gamma$ and contracting indices, we can find the linearized Ricci tensor $R_{\mu \nu }$ and Ricci scalar $R$.

To summarize the results of these calculations:

• Ricci Tensor: $R_{\mu \nu }=\frac{1}{2}({\partial }_{\sigma }{\partial }_{\nu }{h}_{\mu }^{\sigma }+{\partial }_{\sigma }{\partial }_{\mu }{h}_{\nu }^{\sigma }-{\partial }_{\mu }{\partial }_{\nu }h-\square h_{\mu \nu })$
• Ricci Scalar: $R={\partial }_{\mu }{\partial }_{\nu }{h}^{\mu \nu }-\square h$

Here, $h=h_{\mu }^{\mu }={\eta }^{\mu \nu }{h}_{\mu \nu }$ is the trace of the perturbation, and $\square ={\eta }^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }=-{\partial }_t^2+{\nabla }^2$ is the flat-spacetime d'Alembertian operator.

3. The Linearized Einstein Equation

We can now assemble the linearized Einstein Tensor, $G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R$. Again, we use $g_{\mu \nu }\approx {\eta }_{\mu \nu }$ in the second term to remain consistent to first order.
Substituting the expressions for $R_{\mu \nu }$ and $R$ yields a rather complicated expression for $G_{\mu \nu }$.

The linearized Einstein Field Equation is then $G_{\mu \nu }=8\pi G{T}_{\mu \nu }$, where $G_{\mu \nu }$ is the linearized tensor from equation (6.8) in Carroll's notes, and $T_{\mu \nu }$ is the stress-energy tensor, which is treated as a zeroth-order quantity (since the energy density itself must be small to be in the weak-field regime).

As you can see, the equation is a mess. It's a complicated system of second-order partial differential equations for the components of $h_{\mu \nu }$. If this were the end of the story, it would be very difficult to find solutions. Fortunately, it's not.

4. Simplification: The Trace-Reversed Perturbation

There is an algebraic trick that dramatically cleans up the form of the linearized equations. We define a new field, the trace-reversed metric perturbation, ${\bar{h}}_{\mu \nu }$, as:
${\bar{h}}_{\mu \nu }=h_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }h$
This object contains the exact same information as $h_{\mu \nu }$, but it is defined in a way that simplifies the equations. As its name suggests, its trace is the negative of the original trace: $\bar{h}={\eta }^{\mu \nu }{\bar{h}}_{\mu \nu }=h-\frac{1}{2}{\delta }_{\mu }^{\mu }h=h-\frac{1}{2}(4)h=-h$.

With this redefinition, the complicated expression for the Einstein tensor can be shown to simplify significantly.

The Einstein tensor, in terms of ${\bar{h}}_{\mu \nu }$, becomes:
$G_{\mu \nu }=\frac{1}{2}\left({\partial }_{\sigma }{\partial }_{\nu }{\bar{h}}_{\mu }^{\sigma }+{\partial }_{\sigma }{\partial }_{\mu }{\bar{h}}_{\nu }^{\sigma }-{\eta }_{\mu \nu }{\partial }_{\rho }{\partial }_{\sigma }{\bar{h}}^{\rho \sigma }-\square {\bar{h}}_{\mu \nu }\right)$

This is the general form of the linearized Einstein equations. It's still not as simple as a wave equation, because of the derivative terms. The final simplification step does not come from algebra, but from exploiting a physical redundancy in the description: gauge freedom.

Conclusion

In this lesson, we have taken the full, nonlinear theory of General Relativity and successfully derived its linear approximation for weak gravitational fields. This is the foundational step for studying the phenomena of gravitational waves.

Key Takeaways:

• The weak-field limit assumes the metric is a small perturbation on flat spacetime: $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$.
• In this limit, all equations are linearized, meaning terms of order $h^2$ and higher are discarded. This crucially simplifies the Riemann tensor by eliminating the quadratic $\Gamma \Gamma$ terms.
• The resulting linearized Einstein Field Equations are a set of second-order linear partial differential equations for $h_{\mu \nu }$.
• The equations take a more manageable form when written in terms of the trace-reversed perturbation, ${\bar{h}}_{\mu \nu }=h_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }h$.

The expression for $G_{\mu \nu }$ we ended with still looks complicated. However, we have a powerful tool left to use. The choice of coordinates satisfying the weak-field condition is not unique. This freedom, called gauge freedom, is analogous to the gauge freedom of the vector potential in electromagnetism.

Preview of Next Lesson:

In our next lesson, we will exploit this gauge freedom to simplify our equations dramatically. We will introduce a specific gauge choice—the transverse-traceless (TT) gauge—which will make the derivative terms in the linearized Einstein tensor vanish, leaving behind a simple wave equation. Solving this equation will finally reveal the propagating, wavelike nature of gravitational disturbances.