Deriving the Einstein Constant from the Newtonian Limit

The Newtonian Limit: Conditions and Approximations

Newtonian gravity works exceptionally well in a specific domain of the universe: where gravitational fields are weak and objects are moving slowly compared to the speed of light. To recover Newton's laws from General Relativity, we must impose these same conditions.

The Newtonian limit is defined by three key approximations:

1. Weak Gravitational Field: Spacetime is almost flat. This means the metric tensor $g_{\mu \nu }$ can be written as the flat Minkowski metric ${\eta }_{\mu \nu }$ plus a small perturbation, $h_{\mu \nu }$, where the components $|h_{\mu \nu }|\ll 1$.
   $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$
   We will use the $(-,+,+,+)$ signature for the Minkowski metric, so ${\eta }_{00}=-1$. We will neglect any terms of order $h^2$ or higher.
2. Slow Motion: We consider test particles moving at non-relativistic speeds, $v\ll c$. In our units where $c=1$, this means the spatial components of a particle's four-velocity are much smaller than its time component: $|u^i|\ll u^0$.
3. Static Field: The gravitational field is assumed to be time-independent. This implies that the metric components do not change with time: ${\partial }_0{g}_{\mu \nu }=0$.

Our task is to show that under these conditions, the two main equations of GR—the geodesic equation and the Einstein Field Equation—reduce to their Newtonian counterparts:

• Newton's Law of Motion: $\vec{a}=-\nabla \Phi$
• Newton's Field Equation (Poisson's Equation): ${\nabla }^2\Phi =4\pi G\rho$
  where $\Phi$ is the Newtonian gravitational potential and $\rho$ is the mass density.

1. From Geodesics to Newton's Law of Motion

The first step is to see how a particle's trajectory, governed by the geodesic equation, behaves in this limit. The geodesic equation describes the straightest possible path in curved spacetime:
$\frac{d^2{x}^{\mu }}{d{\tau }^2}+{\Gamma }_{\alpha \beta }^{\mu }\frac{d{x}^{\alpha }}{d\tau }\frac{d{x}^{\beta }}{d\tau }=0$
Let's apply our approximations.

Approximating the Geodesic Equation:
The slow-motion condition, $v\ll 1$, means that $d{x}^i/d\tau \ll d{x}^0/d\tau$. Consequently, the sum over $\alpha$ and $\beta$ is overwhelmingly dominated by the term where $\alpha =\beta =0$. The equation simplifies to:
$\frac{d^2{x}^{\mu }}{d{\tau }^2}\approx -{\Gamma }_{00}^{\mu }{\left(\frac{d{x}^0}{d\tau }\right)}^2$

Approximating the Christoffel Symbol:
Next, we calculate ${\Gamma }_{00}^{\mu }$ in our limit. Recall its definition:
${\Gamma }_{00}^{\mu }=\frac{1}{2}{g}^{\mu \lambda }({\partial }_0{g}_{\lambda 0}+{\partial }_0{g}_{0\lambda }-{\partial }_{\lambda }{g}_{00})$

• The static field condition (${\partial }_0{g}_{\mu \nu }=0$) eliminates the first two terms inside the parentheses.
• The weak field approximation allows us to replace the inverse metric with its flat-space version: $g^{\mu \lambda }\approx {\eta }^{\mu \lambda }$.
  This leaves us with:
  ${\Gamma }_{00}^{\mu }\approx -\frac{1}{2}{\eta }^{\mu \lambda }{\partial }_{\lambda }{g}_{00}$

Finding the Equation of Motion:
Let's focus on the spatial components of the acceleration ($\mu =i$). Here, ${\eta }^{i\lambda }$ is only non-zero for $\lambda =i$, where ${\eta }^{ii}=1$. Thus, ${\Gamma }_{00}^i\approx -\frac{1}{2}{\partial }_i{g}_{00}$. The geodesic equation for the spatial coordinates becomes:
$\frac{d^2{x}^i}{d{\tau }^2}\approx \frac{1}{2}{\partial }_i{g}_{00}{\left(\frac{d{x}^0}{d\tau }\right)}^2$
To compare this with Newton's law, we need to re-parameterize the path using coordinate time $t=x^0$ instead of proper time $\tau$. In the slow-motion limit, the relationship is simple: $d\tau \approx dt$. A more precise handling of the chain rule $\frac{d^2{x}^i}{d{t}^2}=\frac{d^2{x}^i}{d{\tau }^2}(\frac{d\tau }{dt}{)}^2+\dots$ shows that to lowest order:
$\frac{d^2{x}^i}{d{t}^2}\approx \frac{d^2{x}^i}{d{\tau }^2}{\left(\frac{dt}{d\tau }\right)}^{-2}\approx \frac{1}{2}{\partial }_i{g}_{00}$
This is the acceleration of a particle as predicted by GR in the Newtonian limit. Comparing this to Newton's law of motion, $\frac{d^2{x}^i}{d{t}^2}=-{\partial }_i\Phi$, we find a direct correspondence:
$\frac{1}{2}{\partial }_i{g}_{00}\approx -{\partial }_i\Phi$
Integrating this gives a relationship between the metric component $g_{00}$ and the Newtonian potential $\Phi$. Since we must recover flat spacetime ($g_{00}={\eta }_{00}=-1$) when gravity is absent ($\Phi =0$), we establish the crucial link:
$g_{00}\approx -1-2\Phi \text{or}{h}_{00}\approx -2\Phi$
This result has a profound physical meaning: in weak gravitational fields, the time-time component of the metric essentially is the Newtonian gravitational potential. Gravitational time dilation, for instance, is directly proportional to $\Phi$.

This entire derivation is performed very clearly in the following video resource.

2. From the EFE to Poisson's Equation

Now we move to the second, and final, part of our task: reducing the Einstein Field Equation to Poisson's equation. This will allow us to fix $\kappa$. It is algebraically simpler to work with the trace-reversed form of the EFE, which you can derive by contracting the standard EFE with $g^{\mu \nu }$ to find $R=-\kappa T$ and substituting back:
$R_{\mu \nu }=\kappa \left(T_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }T\right)$

The Right-Hand Side: The Source

First, let's evaluate the source term in the Newtonian limit. The source is non-relativistic matter (often called "dust"), whose stress-energy tensor is $T^{\mu \nu }=\rho u^{\mu }{u}^{\nu }$, where $\rho$ is the mass density.

• Slow Motion: $u^{\mu }\approx (1,0,0,0)$.
• The only significant component is $T^{00}\approx \rho$.
• The covariant component is $T_{00}=g_{0\alpha }{g}_{0\beta }{T}^{\alpha \beta }\approx (g_{00}{)}^2{T}^{00}\approx (-1{)}^2\rho =\rho$.
• The trace is $T=g^{\mu \nu }{T}_{\mu \nu }\approx g^{00}{T}_{00}\approx {\eta }^{00}{T}_{00}=(-1)\rho =-\rho$.

We are interested in the 00 component of the EFE. Let's plug these into the right-hand side:
$R_{00}=\kappa \left(T_{00}-\frac{1}{2}{g}_{00}T\right)\approx \kappa \left(\rho -\frac{1}{2}(-1)(-\rho )\right)=\kappa \left(\rho -\frac{1}{2}\rho \right)=\frac{1}{2}\kappa \rho$

The Left-Hand Side: The Curvature

Now we compute the Ricci tensor component $R_{00}$ in the weak, static limit.
$R_{00}=R_{0\lambda 0}^{\lambda }={\partial }_{\lambda }{\Gamma }_{00}^{\lambda }-{\partial }_0{\Gamma }_{\lambda 0}^{\lambda }+{\Gamma }_{\lambda \sigma }^{\lambda }{\Gamma }_{00}^{\sigma }-{\Gamma }_{0\sigma }^{\lambda }{\Gamma }_{\lambda 0}^{\sigma }$

• Static Field: The term with ${\partial }_0$ vanishes.
• Weak Field: The terms quadratic in Christoffel symbols (${\Gamma }^2$) are of order $h^2$ and are negligible.
  This leaves us with a greatly simplified expression:
  $R_{00}\approx {\partial }_i{\Gamma }_{00}^i$
  (The sum over $\lambda$ becomes a sum over spatial indices $i$, as ${\Gamma }_{00}^0$ involves a time derivative of $h_{00}$, which is zero).

We already found ${\Gamma }_{00}^i\approx -\frac{1}{2}{\partial }_i{g}_{00}$. Substituting this in:
$R_{00}\approx {\partial }_i\left(-\frac{1}{2}{\partial }_i{g}_{00}\right)=-\frac{1}{2}{\sum }_i{\partial }_i^2{g}_{00}=-\frac{1}{2}{\nabla }^2{g}_{00}$
Finally, we use our result from the geodesic equation, $g_{00}\approx -1-2\Phi$:
$R_{00}\approx -\frac{1}{2}{\nabla }^2(-1-2\Phi )={\nabla }^2\Phi$

Fixing the Constant $\kappa$

We can now equate the left-hand and right-hand sides of the EFE in the Newtonian limit:
$\underset{\text{LHS}}{\underbrace{{\nabla }^2\Phi }}=\underset{\text{RHS}}{\underbrace{\frac{1}{2}\kappa \rho }}$
We compare this directly to Poisson's equation from Newtonian gravity:
${\nabla }^2\Phi =4\pi G\rho$
For the two equations to be identical, we must have:
$\frac{1}{2}\kappa =4\pi G⟹\kappa =8\pi G$
This is the result we were seeking! The unknown coupling constant in Einstein's theory is fixed by the known constant from Newton's theory of gravity.

To see this entire calculation performed, from setting up the approximations to the final result, the following resource is excellent.

Conclusion

In this lesson, we have successfully bridged the gap between General Relativity and classical physics. By analyzing the weak-field, slow-motion limit, we have not only demonstrated the consistency of Einstein's theory but also determined the fundamental coupling constant that relates spacetime geometry to its material content.

Key Takeaways:

• The Newtonian limit is defined by the assumptions of a weak, static gravitational field and non-relativistic motion.
• In this limit, the geodesic equation reduces to Newton's law of motion, $\vec{a}=-\nabla \Phi$, if we make the identification $g_{00}\approx -1-2\Phi$. The Newtonian potential is encoded in the time-time component of the metric.
• The 00 component of the Einstein Field Equation reduces to Poisson's equation, ${\nabla }^2\Phi =4\pi G\rho$.
• Comparing the relativistic and Newtonian field equations fixes the coupling constant to be $\kappa =8\pi G$.
• The complete Einstein Field Equations are therefore:
  $R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R=\frac{8\pi G}{c^4}{T}_{\mu \nu }$
  (Here we have restored the factor of $c^{-4}$ which is necessary for correct units when $c\ne 1$).

Preview of Next Lesson:
We have completed our formulation of the core principles of General Relativity. We now have the EFE, the geodesic equation, and all the necessary constants. The next logical step is to solve these equations for a physical situation. In the next module, we will begin our exploration of the most famous and fundamental solution: the Schwarzschild metric, which describes the spacetime outside a static, spherically symmetric massive object, such as a non-rotating star or black hole. This will be our first concrete application of the full power of the EFE.