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 (usually represented by a vector x indexed by x^μ; μ = 0 … 3 with x⁰ representing the time coordinate and xⁱ; i = 1 … 3 a spatial coordinate vector)

ds² = Σ_μν g_μν dx^μ dx^ν.

Here, g_μν are the coordinates of the metric tensor. 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."

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

and

g =	-c2     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

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

Γ^σ_μν = g^σρ Γ_ρμν

From a given metric, we can compute get the fourth rank space-time curvature tensor R (often called the Reimann tensor, after its inventor)

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. However, our theories of how space-time bends allows us to make predictions about how we could manipulate it if we were able to command such extreme conditions.

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.