Conformal Geometry
Two metrics and are conformally related if for some real function called the conformal factor. (See conformal map).
Looking at the definitions of which tangent vectors are timelike, null and spacelike we see they remain unchanged if we use or As an example suppose is a timelike tangent vector with respect to the metric. This means that . We then have that so is a timelike tangent vector with respect to the too.
It follows from this that the causal structure of a Lorentzian manifold is unaffected by a conformal transformation.
Read more about this topic: Causal Structure
Famous quotes containing the word geometry:
“I am present at the sowing of the seed of the world. With a geometry of sunbeams, the soul lays the foundations of nature.”
—Ralph Waldo Emerson (1803–1882)