John Elton, Retired Chief Scientist at Accusoft. Emeritus Faculty Georgia Tech
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NOTES
Edited by Ed Scheinerman
Indefinite Quadratic Forms
and the Invariance of the Interval
in Special Relativity
John H. Elton
Abstract. In this note, a simple theorem on proportionality of indefinite real quadratic forms
is proved, and is used to clarify the proof of the invariance of the interval in special relativity
from Einstein’s postulate on the universality of the speed of light; students are often rightfully
confused by the incomplete or incorrect proofs given in many texts. The result is illuminated
and generalized using Hilbert’s Nullstellensatz, allowing one form to be a homogeneous poly-
nomial which is not necessarily quadratic. Also a condition for simultaneous diagonalizability
of semi-definite real quadratic forms is given.
1. INTRODUCTION. In the special theory of relativity, an event is a point in space-
time whose coordinates with respect to an inertial reference frame correspond to some
point (t, x, y, z) in R4. Coordinates of events in different inertial reference frames are
assumed to be connected by linear transformations, based on the assumption of homo-
geneity and isotropy of space-time. A famous postulate of Einstein is the universality
of the speed of light: the speed of light in a vacuum is the same in all inertial reference
frames, independent of the motion of the source. One can use the postulate of the uni-
versality of the speed of light, together with the assumption that changes of coordinates
are linear, to determine what changes of coordinates are possible. The idea is to use
this postulate to directly show the invariance of a certain quadratic function of the co-
ordinates, which can in turn be used to determine the linear transformations connecting
the coordinates (called Lorentz transformations). Defining the Lorentz transformations
as the group of linear transformations which leave this quadratic function invariant is
geometrically very appealing. To be most satisfying, and not circular, the invariance of
the quadratic function should be shown to be a simple and immediat