Motion of point particles in a gravitational field.

Ruslan A. Sharipov.

1 ... 14 15 16 ... ru

The action integral for a point particle is the time integral of its Lagrangian.

The Lagrangian itself is simply a function. In the case of a non-baryonic point particle, that is a dark matter particle, it is given by the formula

;

This Lagrangian leads to the following equation of motion for a point particle in a gravitational field, written as a differential equation for the components of the particle's momentum covector:

The formula expressing the components of the particle's momentum covector in terms of its velocity was written out earlier in the text "Lagrangian of Point Particles". Using this formula, we can rewrite the above equation as

It is seen that the particle mass m is completely absent from this equation. In the previous equation, it was explicitly present on the right-hand side and implicitly present on the left-hand side. In the equation derived from it, it has canceled out. This abbreviation is interpreted as a coincidence between inertial and passive gravitational mass. Google AI defines the distinction between active and passive gravitational masses with the following phrase: "Passive gravitational mass is a quantity that characterizes the ability of a body to react to an external gravitational field by experiencing acceleration in response to its action, in contrast to active gravitational mass, which is the mass that creates that field". We will defer a discussion of these two gravitational masses within the framework of the new theory until later.

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