The Metric Tensor
From a projective geometrical perspective, the points within a curvilinear 
dimensional physical spacetime may be viewed as a subset of points, denoted as 
, referred to rectilinear coordinates axes in 
dimensions. Using upper case Roman letters to label the rectilinear coordinate indices, the components of the metric tensor 
of the rectilinear system are constants.
Let 
be the coordinate differential that describes the same vector as 
. The invariant square of the differential interval is given as
![\[d{{s}^{2}}={{g}_{IJ}}d{{x}^{I}}d{{x}^{J}}\]](https://metricofmatter.com/wp-content/ql-cache/quicklatex.com-aabcd36d088b395ca76c9f6f14cf0439_l3.png "Rendered by QuickLaTeX.com")
.
Given the earlier discussion (see the post called “Coordinate Transformations”), the differential can be expressed in terms of the fields in the dimensional spacetime,
![\[d{{x}^{I}}=\sum\nolimits_{f}{\left( \partial {{C}^{I}}/\partial {{f}^{\lambda }}\ \right)}f_{,\mu }^{\lambda }d{{x}^{\mu }}\]](https://metricofmatter.com/wp-content/ql-cache/quicklatex.com-00e5bc691221fbbd5e53af77db99cc68_l3.png "Rendered by QuickLaTeX.com")
which uses the transformation law,
![\[x_{,\mu }^{I}=\sum\nolimits_{f}{\left( \partial {{C}^{I}}/\partial {{f}^{\lambda }}\ \right)}f_{,\mu }^{\lambda }\]](https://metricofmatter.com/wp-content/ql-cache/quicklatex.com-84389a488d97c40943942fdf49dd9563_l3.png "Rendered by QuickLaTeX.com")
.
The squared differential interval is then expressed as
![\[d{{s}^{2}}={{g}_{IJ}}\sum\nolimits_{f}{\left( \partial {{C}^{I}}/\partial {{f}^{\lambda }}\ \right)}f_{,\mu }^{\lambda }d{{x}^{\mu }}\sum\nolimits_{f'}{\left( \partial {{C}^{J}}/\partial {{f}^{\gamma '}}\ \right)}{f}'_{,\nu }^{\gamma }d{{x}^{\nu }}\]](https://metricofmatter.com/wp-content/ql-cache/quicklatex.com-06f6ac5dbd9715a46b566dab40fe2296_l3.png "Rendered by QuickLaTeX.com")
.
As the squared differential interval is form-invariant the metric tensor in the dimensional spacetime may be written as follows:
![\[{{g}_{\mu \nu }}={{g}_{IJ}}\sum\nolimits_{f}{\left( \partial {{C}^{I}}/\partial {{f}^{\lambda }}\ \right)}f_{,\mu }^{\lambda }\sum\nolimits_{f'}{\left( \partial {{C}^{J}}/\partial {{f}^{\gamma '}}\ \right)}{f}'_{,\nu }^{\gamma }\]](https://metricofmatter.com/wp-content/ql-cache/quicklatex.com-2210542095e18f9a232e9f5cbfe89222_l3.png "Rendered by QuickLaTeX.com")
.
In this way, transformation laws that are functionals of the matter fields associated with emergent spacetime points yield a metric tensor that is a functional of the matter fields – the metric of matter.
