Development of generalized metric

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SUMMARY

The discussion centers on the generalization of the spacetime metric in General Relativity (GR) and its potential connection to the kinetic energy term in Newtonian physics. The original poster (OP) posits that the kinetic energy formula, (1/2)m*v^2, can be interpreted as a differential line segment resembling the form n_uv*dx^u*dx^v. This suggests a relationship between kinetic energy and the Riemannian metric on configuration space, as referenced in the provided literature. The OP also hints at the need for incorporating additional time differentials in their equations to further explore this connection.

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We all know how the metric of GR is a generalization of the flat Minkowski spacetime metric. But I wonder if the SR metric is generalized from the kinetic energy term of Newtonian physics. There the kinetic energy is (1/2)m*v^2=(1/2)m*dx*dx. If the mass/2 plays the role of the metric, then this looks like n_uv*dx^u*dx^v, like a differential line segment. Or what part does the kinetic term play in the SR inner product. Thanks.
 
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Wow, that's breaking a butterfly on a wheel. (I hope I got the saying right; is it symptomatic that it involves cannons in german?)
Maybe the OP wants to start with adding an appropriate numder of dt's in his equation:
(1/2)m*v^2=(1/2)m*dx*dx
 

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