Metric Engineering

We live in a geometry that our best theories of reality describe as a combined space-time. The intrinsic curvature of that geometry is what creates the gravity that we experience. This curvature in turn is made by energy (including mass), the movement of energy (including mass), and stress.

Terminology

A given place and time together define an event. The separation of two events is described by an interval s that is a function of the coordinates of the events (in time and space) and a 4×4 array of numbers called the metric tensor, usually denoted with the symbol g.

How the math works: The coordinates are represented by a vector x indexed by xμ; μ = 0 … 3 with x0 representing the time coordinate and xi; i = 1 … 3 a spatial coordinate vector. The coordinates of the metric tensor are given by gμν. The interval is then ds2 = gμν dxμ dxν.

The interval does not change depending on how you define your coordinates (much like the distance between two points does not depend on how you set up your coordinate system to measure the two points) although the components of the metric tensor might. Knowledge of the metric tensor suffices for a full description of the intrinsic curvature of any space-time. Hence, manipulation of space-time is often described as "metric engineering."

Conventions

There are several conventions in the notation of relativity that are worth noting. You can skip this portion if math is not your thing.

There is a distinction between contravariant vectors and covariant vectors. A contravariant vector is the usual "thing with a size pointing in a given direction", such as a displacement or velocity. A covariant vector is a rate at which going forward along the direction changes the value, like a wavenumber. There is a lot deeper mathematical meaning, but for our purposes it suffices to note that a contravariant vector is represented by having its indices in the superscript, x^μ, and a covariant vector by having its indices in the subscript, k_μ. In flat space (or space-time) there is no need to distinguish between these two kinds of vectors; in curved space (or space-time) it becomes necessary. Tensors can have both covariant and contravariant indices.

If an index is repeated, with one a covariant index and the other a contravariant index, you sum over the range of that index from 0 to 3. This is much easier than writing out

ds² = g₀₀ dx⁰ dx⁰ + g₀₁ dx⁰ dx¹ + g₀₂ dx⁰ dx² + … + g₂₃ dx² dx³ + g₃₃ dx³ dx³.

It is also worth noting that there can be sometimes be confusion about whether a superscript indicates a contravariant index or an exponent. The above example, for instance, does not have any exponents, just changing indices. That is, dx² is the 2nd index of dx rather than dx times itself. Usually you can tell because a vector or tensor quantity listed with an index will be referring to the component, not an exponent. From now on, if we need to refer to an exponent and it could be misinterpreted, we will place the quantity to be exponentiated inside parentheses, brackets, or braces to ensure there is no confusion.

For a given metric g_μν, the inverse metric is denoted g^μν and can be found by the normal methods of matrix inversion

g_μσ g^σν = δ^μ_ν

where δ^μ_ν = 1 if μ = ν and equals 0 otherwise.

You can change a covariant vector to a contravariant one (or vice versa) by contracting over the metric tensor

v_μ = g_μν v^ν

v^μ = g^μν v_ν

Example: flat space

The interval between two events at the same time is the distance r: s = r. If x is measured in Cartesian coordinates x = (x, y, z), then we can get the distance from the Pythagorean theorem

r =

√

x2 + y2 + z2

We immediately get that g₁₁ = g₂₂ = g₃₃ = 1 and g_ij = 0 for i ≠ j. In matrix form

g =	1     0     0 0     1 0 0     0 1

Example: flat space-time

In flat space-time, we need to account for the differences in the time coordinate to properly take relativity into account. In this case

s =

√

x2 + y2 + z2 - c2 t2

where c = 299,792,458 m/s is the speed of light. If we define the 0^th component of the distance as c t then

g =	-1     0     0     0 0     1 0 0 0     0 1 0 0     0 0 1

Note that a consequence of this definition is that although a light ray may travel some distance and take some amount of time doing so, the interval of events along the light ray is always zero. We can thus divide events into those with a space-like interval (s > 0), a light-like interval (s = 0), and a time-like interval (s < 0); and these distinctions remain no matter which frame of reference we choose to use in order to analyze the problem.

How to work with a metric

Summary

From the metric, we can determine the curvature of the geometry of space-time. The Einstein field equation directly relates the curvature to the stress-energy tensor. So for a given desired metric and curvature, you can find the necessary stress-energy needed to support that geometry.

There is nothing that requires a given geometry to by physical. Starting with the geometry rather than a physically plausible distribution of stress-energy can lead to strange conditions, such as negative energy densities or material moving faster than light speed. Although these conditions were originally thought to mean a given geometry wasn't possible, this has later been questioned by more recent investigators. Nor is it required for the resulting stress-energy to obey the continuity equations for the motion of energy with time; whereas any physical distribution of energy would need to obey these continuity relations.

Concepts

Before covering how to convert a given geometry, described by a metric, into stresses and flows and energy densities, let's cover two basic ideas for talking about geometries. The math for how to find these is a bit more in depth than this article will handle, but you can find it in many standard references.

First is the idea of the straightest possible lines in a curved geometry. Obviously, a curved geometry can't really have straight lines. But the lines that curve the least are useful for doing a lot of things. These are called geodesics. As an example, the geodesics on the surface of a sphere (like a globe) are the great circles (such as the equator or lines of longitude – but not lines of lattitude!). In relativity, objects in free fall move along geodesics of the combined space-time geometry.

The other idea is that if you have a vector and you have a curve on a given geometry (which can be a geodesic but which does not have to be) you can move the vector along the curve in such a way that it maintains its angle and orientation with the curve. This is called parallel transport.

Working it all out

Here's where we get into the nitty-gritty math. Feel free to skip this section if you think math is icky.

From the metric, we can get the Christoffel symbols Γ^σ_μν

Γ_σμν = [∂_ν g_σμ + ∂_μ g_σν - ∂_σ g_μν]/2

Γ^σ_μν = g^σρ Γ_ρμν

Here, ∂_μ is the partial derivative with respect to the coordinate μ.

Interpretation: In a curved coordinate system – such as those in a curved geometry, but also curved coordinate systems in a flat geometry such as cylindrical or spherical polar coordinate systems – if you move to a different place the directions your coordinate system point in change. The Christoffel symbol, also called the connection coefficient, is the rate at which a basis vector εμ for your coordinate system changes as it is moved in a given direction ∂νεμ = Γσμν εσ

From the Christoffel symbols, we can find the fourth rank space-time curvature tensor R^ρ_σμν (often called the Reimann tensor, after its inventor)

R^ρ_σμν = ∂_μ Γ^ρ_νσ - ∂_ν Γ^ρ_μσ + Γ^ρ_μλ Γ^λ_νσ - Γ^ρ_νλ Γ^λ_μσ

R_ρσμν = g_ρλ R^λ_σμν

Interpretations: One interpretation of the Reimann tensor is that if you have closed infinitesimal parallelogram with one set of sides given by the vector u and the other set of sides given by the vector v; then a vector a transported around the parallelogram, maintaining its orientation with respect to the translation along each side will, when it has returned, have changed by an amount Δaρ = -Rρσμν aσ uμ vν Another interpretation is that the Reimann tensor describes tidal accelerations. If you have two objects at rest with respect to each other separated by a distance b and if the objects have a four-velocity u, then the tidal acceleration is D2bρ = -Rρσμν uσ bμ uν. dτ2 In particular, at rest with respect to the objects the four velocity is u0 = c and ui = 0, i = 1, 2, 3, such that (if you are using a Cartesian coordinate system) d2bi = -c2 Ri0j0 bj. dt2

With the Riemann tensor, you can find a second rank curvature tensor named the Ricci tensor, R_σμ

R_σμ = g^ρν R_ρσμν

and the scalar curvature R

R = g^σμ R_σμ

(there seems to be a trend toward the over-use of R as a symbol).

Interpretations: The Ricci tensor gives information about how volume in your geometry changes. If you have a volume V and move that volume a small distance a along a set of initially parallel geodesics in the direction x (this is as close as we can get to rigidly moving the volume in one direction in a flat geometry by the distance a), the volume will change by δV = -a2 Rμν xμ xν V The scalar curvature gives a coordinate-independent averaged measure of the curvature. A positive R indicates that the geometry is locally spherical, a negative R is a geometry that is locally hyperbolic. It also describes how volume and area (area and perimeter for 2D geometries) changes in curved coordinate systems. For a geometry in n dimensions, a small ball of radius r with Cartesian volume VC and area AC is distorted such that V ≈ [1 -   R   r2 ] VC 6 (n + 2) A ≈ [1 -   R   r2 ] AC 6 n A circle of a given radius has a smaller circumference and area in a spherical geometry than a flat geometry. The circle in flat 2D space is shown with a dashed perimeter and radius, the circle in a spherical 2D geometry is shown with a solid perimeter and radii. A circle of a given radius has a larger circumference and area in a hyperbolic geometry than a flat geometry. The circle in flat 2D space is shown with a dashed perimeter and radius, the circle in a hyperbolic 2D geometry is shown with a solid perimeter and radii.

Now we can relate the curvature to the stress-energy T_μν using the Einstein field equation

Rμν -	1	R gμν =	8 π G	Tμν
	2		c4

G = 6.6743×10⁻¹¹ m³/kg/s² is the gravitational constant. π ≈ 3.14159 is the circle constant. T₀₀ is the energy density of the space-time. This term usually dominates the stress-energy tensor and usually the contribution of matter with an energy of E = m c² for mass m dominates the energy. T_0i = T_i0, i = 1 … 3, is the i component of the momentum density. T_ij = T_ji, i, j = 1 … 3, is the ij component of the Cauchy stress tensor, describing the pressures, tensions, and shears (collectively, stress) in that region of space-time. In particular, for a unit area with a normal (vector perpendicular to the area face) in direction i, T_ij is the j component of the force acting across that surface.

Thus, for a given metric, we can work out the stress-energy needed to support that metric.

Terminology: The tensor Gμν = Rμν - (1/2) R gμν is called the Einstein tensor. Thus, the Einstein field equations may be made more compact as Gμν = (8 π G / c4) Tμν. Interpretations: The Einstein tensor gives information about the local curvature of three-dimensional space. The scalar curvature of the 3-dimensional space orthogonal to a given time-like direction t is R3D = -2 Gμν tμ tν. The energy density in the frame of reference where t has no spatial components at all (it points purely forward in time) is ρ = Tμν tμ tν. Thus, we can relate the scalar curvature of space directly to the energy density R3D = -(4 π G / c4) ρ.

Practical details

Engineering the curvature of space-time is currently (2026) far beyond our reach; requiring energies, stresses, or mass flows far beyond anything we have managed to create.

Examples of metric engineering

Black holes are an extreme form of object defined by their intense space-time curvature. The page on black hole engineering describes some of the things we might do if we could make or otherwise get our hands on one of these objects.

Wormholes are shortcuts between to locations in space-time. They are entirely hypothetical, but if they did exist the page on wormholes describes what we know about their properties.

Warp drives are hypothetical methods of moving faster than light. There is still much to be learned about these solutions to general relativity, the page on warp drives goes over some of what we do know.

Credit

Author: Luke Campbell