# Is mass dual to a metric?

1. Jun 26, 2008

### friend

I wonder if the mass term in the kinetic energy can be alterantively interpreted as a metric? And can energy be a form of inner product?

I've found a few references that suggest these things, but I'm not sure how accepted they are. And I'm wondering if anyone can provide any better insight into all this.

Frankel, 1997, The Geometry of Physics, page 50 and pages 54-55, describes the kinetic energy term of the lagrangian as the inner product of a generalized velocity contravariant vector in the tangent bundle of the configuration space with the generalized momentum covector in the cotangent bundle (or phase space) of the same configuration space. He writes that the kinetic energy, T, is frequently a positive definite symmetric quadratic form in the velocities. Equation (2.31), pg 55

$$$T\left( {q,\dot q} \right) = \frac{1}{2}\sum\limits_{jk} {g_{jk} (q)\dot q^j \dot q^k }$$$

which looks a lot like the kinetic energy of a particle

$$$T = {\textstyle{1 \over 2}}m\dot q\dot q$$$

where the mass, m, plays the role of the metric g(q), and the kinetic energy is a type of inner product.

Also, in Robert Wald, 1984, General Relativity, page 61, the "energy-momentum" 4-vector is $$$p^a = m \cdot u^a$$$ , where $$$u^a$$$ is the 4-velocity of a particle. And the "energy" of a particle as observed by an observer with 4-velocity, $$$v^a$$$ , is $$$E = -mu_a \cdot v^a$$$. Again this looks like an inner product where m plays the role of a metric. Though I have not found whether there is a special name given to the tangent bundle or cotangent bundle of the Minkowski spacetime.

And in Kleinert, 2006, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th Edition, page 767, he simply transforms the Action integral of the flat space kinetic energy,

$$${\rm A} = \int_{t_a }^{t_b } {dt{\textstyle{M \over 2}}\,(\dot x^i )^2 \,}$$$

to the curved space version,

$$${\rm A} = \int_{t_a }^{t_b } {dt{\textstyle{M \over 2}}g_{\mu \nu } (q)\,} \dot q^\mu \dot q^\nu$$$

which again looks like an inner product between vectors and covectors, and where M/2 can be seen as a scale factor that can be incorporated into $$${g_{\mu \nu } (q)}$$$.

So my question is how well established is this dual relationship between the mass and the metric? Did I stumble onto some vauge references? Or is this well established in the literature? And what would this mean for quantum gravity concerns? Is it fair to talk about coupling mass to quantum geometry of QG when the metric is a dual to mass? Thanks.

Last edited: Jun 26, 2008