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Relativistic invariance (or Lorentz invariance) means "the same regardless of frame of reference". For example, a relativistic invariant quantity would be the same if you measured it while you are at rest versus if you measured it while you are moving at a constant velocity. It applies to invariances that hold even if the relative velocity between the frames of reference is relativistic or ultrarelativistic. If the relativistic-invariant quantity requires the Lorentz transform when shifting frames of reference, its relativistic invariance indicates the transform's output will be the same as its input. Sometimes the phrase relativistic invariance is used specifically to mean such a case.
An example is the speed of light (c), and the discovery of its invariance (through measurement) was a prime motivator for the development of relativity and the Lorentz transform. Applying the transform to the speed of light does indeed always yield the same value. Mass (as the word is generally used, i.e., as the "rest mass") is invariant, but that is because, as defined, one doesn't apply the Lorentz transform.
The terms are also used for laws of physics, i.e., equations, that remain true in different frames of reference. Some familiar laws are invariant, and some have specific relativistic versions that are invariant, i.e., that show their relationship using only invariant quantities, and some are "meta", e.g., the laws of relativity itself.
Lengths and time intervals are famously not invariant: a foot-long ruler no longer measures to exactly a foot when measured from a frame of reference in which the ruler is moving. A spacetime interval (for two events happening some distance and time interval apart, it is square root of the difference between the distance squared and the time interval squared) does remain unchanged under the Lorentz transform. Spacetime's minimum time interval or minimum distance between two events is also a fixed number, i.e., the smallest of all possible frames of references' the time intervals or distances between the two events.