Ricci Tensor in Riemann Normal Coordinates

Recall that, in the context of Newtonian gravity, if we consider an isolated spherical shell of radius r of dust particles initially at rest surrounding a distribution of matter of uniform density ρ, the volume within the shell is V(r) = (4/3)πr³ and the enclosed mass is m = ρV. Each particle of the shell will undergo inward acceleration according to d²r/dt² = –m/r² (in units where Newton’s gravitational constant is 1). The derivatives of the volume are

Therefore, at the initial moment, when dr/dt = 0, we have

and hence we have

To find the corresponding relationship in general relativity, consider Einstein’s field equations written (with G and c equal to 1) in the form

where R_μν is the Ricci tensor and T_μν is the energy-momentum tensor. At the origin of normal coordinates the metric tensor is the diagonal Minkowski metric with components 1, –1, –1, –1, and for a spherical uniform distribution of matter, with no transverse stresses, at rest in these coordinates, the only non-zero components of the energy-momentum tensor are the mass-energy density T_tt = ρ and the pressures T_xx = P_x, T_yy = P_y, and T_zz = P_z. Under these conditions and for this system of coordinates, the trace is T = T_tt – T_xx – T_yy – T_zz and the “tt” equation of (2) is

Aside from the appearance of the pressures, this would match the Newtonian equation (1) if we could show that R_tt equals the second derivative of the volume divided by the volume (or, equivalently, the second derivative of the logarithm of the volume).

In this context each particle of dust in the spherical shell is following a geodesic path, initially stationary in the local rest frame of the center of the sphere. Letting σ denote the vector from the center to one of the dust particles (located an incremental distance away), normal to the path of the central point, the equation of geodesic deviation is expressed in terms of a second absolute (covariant) derivative

where v is the vector along the path of the central point, Rⁿ_abc is the mixed Riemann curvature tensor, and τ is a path parameter. The symbol δ refers to the absolute (covariant) derivative, although the distinction between this and the ordinary total derivative at the origin of normal coordinates is rarely emphasized. The situation is illustrated in the figure below.

In our case the center of the sphere is stationary, so it has components v⁰ = 1 and v₁ = v₂ = v₃ = 0. Thus the only non-vanishing terms are given by

The second derivative of the volume can be expressed in terms of the average of the second derivatives of the radial positions for dust particles in four orthogonal basis directions (as can be shown by considering the volume V=wxyz of a small four-dimensional “box” with edge lengths w, x, y, and z, initially equal and accelerating from rest), so we consider the four deviation vectors

where r is an increment length representing the radius of the shell. Thus we have

The expression in the square brackets is the –R₀₀ component of the Ricci tensor (noting our sign convention for the contraction of the Riemann tensor), so we can write

The volume and its derivatives of a 4-dimensional sphere of radius r in a flat manifold are

At the initial time the first derivatives vanish, and we have

Identifying the average value of the second absolute derivative of the radial vector σ with the second derivative of the radius r, we arrive at

and hence equation (3) is sometimes expressed as

This expression has appeared in the literature (usually with covariant derivatives), such as in Section 19.7 of Penrose (2004). Penrose cites equation (2), and then says “the inward acceleration of the volume” is 4π(ρ + P_x + P_y + P_z).  He doesn’t say that the second derivative is divided by the volume, but this might be glossed over in the word “acceleration”, perhaps implicitly referring to logarithmic acceleration.

For a closely-related result, in another article we noted that, at the origin of Riemann normal coordinates, the components of the Ricci tensor (contraction of the Riemann curvature tensor) can be expressed as

where g is the determinant of the covariant metric tensor g_μν. Making use of (3), this implies for free-falling normal coordinates

The square root of the metric determinant is the volumetric scale factor for the n-dimensional space(time), i.e., for the volume V we have

In view of this, it would be tempting to think that the second derivative of this volumetric scale factor, divided by the scale factor, must equal the second derivative of the volume (of a geodesic sphere) divided by that volume. However, these are actually two different things. The former involves the volume within a region whose boundaries are constant coordinates, whereas the latter involves the volume with a region whose boundaries are formed by geodesic particles. These volumes need not have the same second derivatives. Indeed, the right side of equation (6) has an extra factor of 1/3 compared with equation (4). This comes directly from the factor of 3 in equation (5).  We will confirm this factor below.

Incidentally, Section 5.2 of Wald (1984) gives the equation for the scale factor a(τ) in an isotropic and homogeneous cosmology

which includes a factor of 1/3, although the cosmological scale factor is distinct from the volumetric scale factor. The cosmological  scale factor is a linear scale, so the volume would correspond to the cube of this quantity, and hence account for the factor of 1/3, noting that ln(a³) = 3ln(a). But the quantity √-g is not the cube root of the volume.

As another aside, Feynman’s lectures included his famous recipe: For a spherical region with surface area A and radius r the Einstein field equations imply

where M is the mass within the sphere, and ρ is the density. This is admittedly a very different relation, referring to the “radius excess” instead of the volume (or scale) acceleration, but the factor of 1/3 might still have some relevance. In any case, we now return to the task of verifying that the factor of 3 in equation (5) is correct. We will do this in three different ways.

First, let’s review the original reasoning, leading to the factor of 3, which is fairly straight-forward. The Christoffel symbols all vanish at the origin of Riemann normal coordinates, so the Ricci tensor is

where ,n denotes partial differentiation with respect to xⁿ. Making use of the cyclic skew symmetry

we can substitute for the second term on the right side of the preceding expression to give

We also have

and since the first derivatives of g are zero at the origin of Riemann normal coordinates, and since partial differentiation is commutative, it follows that all three of the terms are equal, and we arrive at equation (5).

To confirm this in a second way, consider the simple example (from Section 5.7 of Reflections on Relativity) of a two dimensional surface whose height above the XY plane (where X and Y are Cartesian coordinates) is h = bXY. By the transformation X = x/(1 + uy²) and Y = y/(1 + ux²) where u = b²/3 the x,y coordinates are Riemann normal coordinates on this surface, and the covariant components of the metric tensor are

with the determinant g = 1 + u(x² + y²). As expected, the first partial derivatives of g vanish at the origin. The contravariant components of the metric tensor are

The only non-zero Christoffel symbols of the first kind are

and consequently the Christoffel symbols of the second kind are

This confirms that the Christoffel symbols all vanish at the origin. Now we can evaluate the xx component (for example) of the Ricci tensor directly using equation (5) as follows:

Thus at the origin we have R_xx = 3u. Now, using the alternative expression (1) for the components of the Ricci tensor (at the origin of Riemann normal coordinates), we have

which again evaluates to R_xx = 3u at the origin. Therefore, the factors of 3 in equations (5) seems to be confirmed.

A third approach is to consider the expansion of the metric tensor

where the metric expressions on the right side are evaluated at the point p. As discussed elsewhere, we have the cyclic skew symmetries

and we also have, at the origin of normal coordinates, the relation

which we can write equivalently in either of the two other forms with permuted indices

Combining these with the cyclic symmetry we can solve for the g derivatives, yielding

Now, if we consider the expansion of the metric tensor at the origin of Riemann normal coordinates, the first derivatives of the metric components vanish by definition, so the lowest-order non-zero term is the second-degree term

which of course represents a summation over the indices α and β. Making use of the preceding relation, the terms with α = β can be written as

For the remaining term, with α ≠ β, we can combine the coefficients of x^αx^β and x^βx^α in the summation, so we have

Thus, accounting for all the terms of the summation, we have at the origin (p=0) of Riemann normal coordinates (where the metric tensor is just the Minkowski metric η_μν)

Recalling that R_μανβ = g_μσR^σ_ανβ, and that the Ricci tensor is defined as R_αβ = R^ν_αβν (noting the sign convention with this definition, and the two permuted indices), this can be written as

Taking the determinants of both sides, and then taking the square roots, we have

Taking partial derivatives with respect to xa and xb (nothing that we get a factor of two both for the diagonal terms and the off-diagonal terms since the latter have two symmetrical components), we have at the origin of Riemann normal coordinates

This again is consistent with the factor of 3 in equation (5).

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