Metric Engineering - Revision history

Lwcamp at 19:04, 7 March 2026

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	==References==		==References==

			[[Category:Physics & Math & Engineering]]

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Summary

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	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.		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.		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===		===Concepts===

Lwcamp: /* Terminology */

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Terminology

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	==Terminology==		==Terminology==

	A given place and time together define an <i>event</i>. The separation of two events is described by an interval <i>s</i> that is a function of the coordinates (usually represented by a vector <i><b>x</b></i> indexed by <i>x</i><sup>μ</sup>; μ = 0 … 3 with <i>x</i><sup>0</sup> representing the time coordinate and <i>x</i><sup>i</sup>; i = 1 … 3 a spatial coordinate vector)<ref name="Arfken">G. B. Arfken and H. J. Weber, "Mathematical Methods for Physicists, Fourth Edition", Academic Press, San Diego (1995)</ref>		A given place and time together define an <i>event</i>. The separation of two events is described by an interval <i>s</i> that is a function of the coordinates of the events (in time and space) and a 4×4 array of numbers called the <i>metric tensor</i>, usually denoted with the symbol <i>g</i>.
			<table border=1>
			<tr><td>How the math works: The coordinates are represented by a vector <i><b>x</b></i> indexed by <i>x</i><sup>μ</sup>; μ = 0 … 3 with <i>x</i><sup>0</sup> representing the time coordinate and <i>x</i><sup>i</sup>; i = 1 … 3 a spatial coordinate vector. The coordinates of the metric tensor are given by <i>g</i><sub>μν</sub>. The interval is then<ref name="Arfken">G. B. Arfken and H. J. Weber, "Mathematical Methods for Physicists, Fourth Edition", Academic Press, San Diego (1995)</ref>
	<div align=center>		<div align=center>
	<i>ds</i><sup>2</sup> = <i>g</i><sub>μν</sub> <i>dx</i><sup>μ</sup> <i>dx</i><sup>ν</sup>.		<i>ds</i><sup>2</sup> = <i>g</i><sub>μν</sub> <i>dx</i><sup>μ</sup> <i>dx</i><sup>ν</sup>.
	</div>		</div>
	~~Here, <i>g</i><sub>μν</sub> are the coordinates of the <i>metric tensor~~</i>.		</table>
	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.		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."		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."

Lwcamp at 19:10, 4 March 2026

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	~~{{PageConstructionNotice}}~~

	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.		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.

Lwcamp at 19:10, 4 March 2026

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	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\|warp drives]] goes over some of what we do know.		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\|warp drives]] goes over some of what we do know.

			==Credit==
			Author: Luke Campbell

	==References==		==References==

Lwcamp: /* Working it all out */

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Working it all out

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	<tr><td> Terminology: The tensor <i>G</i><sub>μν</sub> = <i>R</i><sub>μν</sub> - (1/2) <i>R</i> <i>g</i><sub>μν</sub> is called the Einstein tensor. Thus, the Einstein field equations may be made more compact as <i>G</i><sub>μν</sub> = (8 π <i>G</i> / <i>c</i><sup>4</sup>) <i>T</i><sub>μν</sub>.		<tr><td> Terminology: The tensor <i>G</i><sub>μν</sub> = <i>R</i><sub>μν</sub> - (1/2) <i>R</i> <i>g</i><sub>μν</sub> is called the Einstein tensor. Thus, the Einstein field equations may be made more compact as <i>G</i><sub>μν</sub> = (8 π <i>G</i> / <i>c</i><sup>4</sup>) <i>T</i><sub>μν</sub>.

	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 <i><b>t</b></i> is<ref name="Loveridge"></ref> <i>R</i><sub>3D</sub> = -2 <i>G</i><sub>μν</sub> <i>t</i><sup>μ</sup> <i>t</i><sup>ν</sup>.		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 <i><b>t</b></i> is<ref name="Loveridge"></ref> <i>R</i><sub>3D</sub> = -2 <i>G</i><sub>μν</sub> <i>t</i><sup>μ</sup> <i>t</i><sup>ν</sup>. The energy density in the frame of reference where <i><b>t</b></i> has no spatial components at all (it points purely forward in time) is ρ = <i>T</i><sub>μν</sub> <i>t</i><sup>μ</sup> <i>t</i><sup>ν</sup>. Thus, we can relate the scalar curvature of space directly to the energy density <i>R</i><sub>3D</sub> = -(4 π <i>G</i> / <i>c</i><sup>4</sup>) ρ.
	</table>		</table>

Lwcamp: /* How to work with a metric */

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How to work with a metric

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

			<table border=1>
			<tr><td> Terminology: The tensor <i>G</i><sub>μν</sub> = <i>R</i><sub>μν</sub> - (1/2) <i>R</i> <i>g</i><sub>μν</sub> is called the Einstein tensor. Thus, the Einstein field equations may be made more compact as <i>G</i><sub>μν</sub> = (8 π <i>G</i> / <i>c</i><sup>4</sup>) <i>T</i><sub>μν</sub>.

			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 <i><b>t</b></i> is<ref name="Loveridge"></ref> <i>R</i><sub>3D</sub> = -2 <i>G</i><sub>μν</sub> <i>t</i><sup>μ</sup> <i>t</i><sup>ν</sup>.
			</table>

	==Practical details==		==Practical details==

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	</div>		</div>

	The scalar curvature gives a coordinate-independent averaged measure of the curvature. A positive <i>R</i> indicates that the geometry is locally spherical, a negative <i>R</i> is a geometry that is locally hyperbolic. It also describes how volume and area changes in curved coordinate systems. For a geometry in <i>n</i> dimensions, a small ball of radius <i>r</i> with Cartesian volume <i>V</i><sub>C</sub> and area <i>A</i><sub>C</sub> is distorted such that<ref name="Loveridge"></ref>		The scalar curvature gives a coordinate-independent averaged measure of the curvature. A positive <i>R</i> indicates that the geometry is locally spherical, a negative <i>R</i> 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 <i>n</i> dimensions, a small ball of radius <i>r</i> with Cartesian volume <i>V</i><sub>C</sub> and area <i>A</i><sub>C</sub> is distorted such that<ref name="Loveridge"></ref>
	<div align=center>		<div align=center>
	<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >		<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >

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Working it all out

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	<table align=center>		<table align=center>
	<tr><td width=300>[[File:Curvature_spherical.png]] <td width=300>[[File:Curvature_hyperbolic.png]]		<tr><td width=300>[[File:Curvature_spherical.png]] <td width=300>[[File:Curvature_hyperbolic.png]]
	<tr><td>A circle of a given radius has a smaller circumference and area in a spherical geometry than a flat geometry. <td>A circle of a given radius has a larger circumference and area in a hyperbolic geometry than a flat geometry.		<tr><td>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. <td>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.
	</table>		</table>
	</table>		</table>

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Working it all out

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	</table>		</table>
	</div>		</div>
			<table align=center>
			<tr><td width=300>[[File:Curvature_spherical.png]] <td width=300>[[File:Curvature_hyperbolic.png]]
			<tr><td>A circle of a given radius has a smaller circumference and area in a spherical geometry than a flat geometry. <td>A circle of a given radius has a larger circumference and area in a hyperbolic geometry than a flat geometry.
			</table>
	</table>		</table>