isometry love

# isometry

## Definitions

### from The American Heritage® Dictionary of the English Language, 5th Edition.

• noun Equality of measure.
• noun Equality of elevation above sea level.
• noun Mathematics A function between metric spaces which preserves distances, such as a rotation or translation in a plane.
• noun Biology A proportional change in the size of a part or parts of an organism as the organism grows.

• noun mathematics A function between metric spaces (or on a single metric space) having the property that the distance between two images is equal to the distance between their pre-images.

• noun equality of elevation above sea level
• noun equality of measure (e.g., equality of height above sea level or equality of loudness etc.)
• noun the growth rates in different parts of a growing organism are the same
• noun a one-to-one mapping of one metric space into another metric space that preserves the distances between each pair of points

## Etymologies

From iso- and -metry.

## Examples

• The metric has time dependent metric elements, which means there is no Killing isometry which defines energy conservation as a symmetry of the spacetime.

• Temperature-mediated transitions between isometry and allometry in a colonial, modular invertebarte.

• Further momentum is an isometry of this spacetime, so momentum is conserved.

Does Space Expand? Sean 2008

• The metric coefficients are time dependent, and this precludes the existence of a Killing vector K_t that defines an isometry and energy conservation law.

Being a Heretic is Hard Work Sean 2008

• In effect cosmologies in their lack of such isometry permits the creation of momentum and energy — aka the big bang.

• With a few [extremely non-generic] exceptions, namely the ones with maximal isometry groups, one can say definitely whether they are expanding or not.

Does Space Expand? Sean 2008

• This process of being dragged along is due to a symmetry (isometry) of space which preserves momentum.

Be vewwwwy vewwwwy quiet…. Julianne 2008

• (M², g²ab), respectively, and if there exists an isometry

Time Machines Earman, John 2004

• Does it follow, as we would want determinism to guarantee, that Î¨ is extendible to an isometry from

Time Machines Earman, John 2004

• (M*, g*ab) violates the condition that determinism was supposed to guarantee as Î¨ is not extendible to an isometry from D+ (Î£) onto

Time Machines Earman, John 2004