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Lorentzian Inner Product


The standard Lorentzian inner product on R^4 is given by

 -dx_0^2+dx_1^2+dx_2^2+dx_3^2,
(1)

i.e., for vectors v=(v_0,v_1,v_2,v_3) and w=(w_0,w_1,w_2,w_3),

 <v,w>=-v_0w_0+v_1w_1+v_2w_2+v_3w_3.
(2)

R^4 endowed with the metric tensor induced by the above Lorentzian inner product is known as Minkowski space and is denoted R^(1,3).

The Lorentzian inner product on R^4 is nothing more than a specific case of the more general Lorentzian inner product <·,·> on n-dimensional Lorentzian space with metric signature (1,n-1): In this more general environment, the inner product of two vectors x=(x_0,x_1,...,x_(n-1)) and y=(y_0,y_1,...,y_(n-1)) has the form

 <x,y>=-x_0y_0+x_1y_1+...+x_(n-1)y_(n-1).
(3)

The Lorentzian inner product of two such vectors is sometimes denoted x degreesy to avoid the possible confusion of the angled brackets with the standard Euclidean inner product (Ratcliffe 2006). Analogous presentations can be made if the equivalent metric signature (n-1,1) (i.e., (3,1) for Minkowski space) is used.

The four-dimensional Lorentzian inner product is used as a tool in special relativity, namely as a measurement which is independent of reference frame and which replaces the typical Euclidean notion of distance. For a four-vector x=(x_0,x_1,x_2,x_3) in Minkowski space, the variables x_1, x_2, and x_3 can be thought of as space variables with x_0 as the time variable. In various literature, the time variable is sometimes labeled t; moreover, when used in general relativity, either of x_0=ct or x_0=ict may be used where c denotes the speed of light and where i=sqrt(-1) denotes the imaginary unit (Misner et al. 1973). For simplicity, the formula (2) uses the conventions of real time coordinates and appropriately-chosen units so that the speed of light has the value c=1.

For a vector v, the sign of v^2=<v,v> determines the type of v: In particular, if v^2>0, then v is spacelike; if v^2=0, then v is lightlike; and if v^2<0, then v is called timelike. After a change of variables, it is possible to rewrite the Lorentzian inner product as above where t is in the direction of a given timelike vector v with <v,v>=-1. Such a change of variables corresponds to a change in reference frame. Collectively, the set of all reference frame changes form the Lorentz group, also called the orthogonal group O(1,3) (or O(3,1) when using the (3,1) metric signature).


See also

Inner Product, Lightlike, Lorentz Group, Lorentzian Space, Metric Signature, Metric Tensor, Minkowski Space, Orthogonal Group, Spacelike, Timelike

Portions of this entry contributed by Todd Rowland

Portions of this entry contributed by Christopher Stover

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References

Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco, CA: W. H. Freeman, p. 53, 1973.Ratcliffe, J. G. Foundations of Hyperbolic Manifolds. New York: Springer-Verlag, 2006.

Referenced on Wolfram|Alpha

Lorentzian Inner Product

Cite this as:

Rowland, Todd; Stover, Christopher; and Weisstein, Eric W. "Lorentzian Inner Product." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LorentzianInnerProduct.html

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