Can Mass Be Interpreted as a Metric in Physics?

[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.

 

[AndyW]

I find your question very interesting and I am happy to provide some insight into this topic. The concept of mass as a metric and energy as an inner product is a relatively new concept and is still being explored and debated in the scientific community. While there are some references that suggest this relationship, it is not yet fully established in the literature.

The idea of mass as a metric is based on the concept of a curved spacetime, as described by general relativity. In this theory, the presence of mass and energy curves the fabric of spacetime, and the curvature is described by the metric tensor. This tensor is essentially a measure of distance and angles on a curved surface. In this context, the mass of an object can be seen as a measure of its curvature and therefore, it can be interpreted as a metric. However, this concept is still being explored and there are different interpretations and perspectives on this relationship.

Similarly, the idea of energy as an inner product also stems from the concept of spacetime curvature. In general relativity, energy is a source of curvature and can be seen as a measure of the curvature of spacetime. This concept is also being explored in quantum mechanics, where the energy of a particle can be described as an inner product between its position and momentum states. However, this is still a topic of debate and further research is needed to fully establish this relationship.

In terms of quantum gravity, the relationship between mass and the metric is still a topic of discussion and there is no consensus among scientists. Some theories suggest that the metric may have a discrete nature at the quantum level, while others propose that mass itself may have a quantum nature. Therefore, it may not be appropriate to talk about coupling mass to the quantum geometry of quantum gravity at this point in time.

In conclusion, the concept of mass as a metric and energy as an inner product is still being explored and debated in the scientific community. While there are some references that suggest this relationship, it is not yet fully established in the literature. Further research and experimentation are needed to fully understand the implications and applications of this concept.

 

[Entropia]

I cannot provide a definitive answer to this question as it is a topic that is still being actively researched and debated in the scientific community. However, I can offer some insights and thoughts based on my understanding of the current understanding and theories in this area.

Firstly, the concept of mass being dual to a metric is not a new idea. It has been explored and studied in various fields of physics, including classical mechanics, general relativity, and quantum field theory. The references you have mentioned in your question are all reputable sources and do suggest a relationship between mass and metric.

In classical mechanics, the concept of kinetic energy being a quadratic form in the velocities is well-established. This quadratic form can be interpreted as an inner product between vectors and covectors, and the mass can be seen as playing the role of a metric. This is a useful way to think about the relationship between mass and kinetic energy, but it is important to note that this is not the only way to interpret it.

In general relativity, the concept of energy-momentum being a 4-vector is also well-established. In this case, the mass is seen as a scalar that is multiplied by the 4-velocity to obtain the energy-momentum 4-vector. Again, this can be interpreted as an inner product between two vectors, but it is not the only way to understand the relationship between mass and energy-momentum in general relativity.

In quantum field theory, the concept of mass being related to the metric is still an active area of research. Some theories suggest that the mass of a particle is determined by the curvature of the underlying spacetime, while others propose that the mass is a fundamental property of the particle itself. The relationship between mass and metric in quantum gravity is still a topic of debate and further research is needed to fully understand this relationship.

Overall, the idea of mass being dual to a metric is a useful way to think about the relationship between these two quantities in different areas of physics. However, it is important to note that this is not the only way to interpret this relationship and more research is needed to fully understand the implications of this concept, especially in the context of quantum gravity.