Physical Foundations of Natural Science Vasily Beskin # 2-4

Gravitation and Astrophysics

School level Ready solutions of equations (many of which are not even formulated)

An example - Gravity What is the equation and that - the solution?

An example - Gravity What is the equation and that - the solution?

Scientific level The formulation of equations and their solution

A fundamental observation I.Newton ( ) Our world is described by equations of the second order Requires two initial conditions

Classiacal world (c, G) Fundamental generalizations Mechanics (Newton laws): Lorentz invariance, conservation laws are a consequence of the symmetry, integrated nature conservation laws, principle of least action (Lagrange) Hamiltonian formalism, from scalars to tensors.

Principle of least action W.Hamilton ( ) J.L.Lagrange ( ) coordinate x momentum p = m v velocity v

Principle of least action P.de Fermat ( ) P.L.de Maupertuis( ) L.Euler ( )

Principle of least action Fermat's principle x 1 x 2 x у

Principle of least action Action Equation of motion

Principle of least action The equations of motion can be derived from the principle of least (extreme) action.

Newton limit From school we know that

An example – the Gravity

Scalar potential Acceleration does not dependent on the mass The law of motion looks identical Right to left and left to right The theory of gravity is scalar

Certainly, the correct theory Prediction of the existence of planets Correctly describes the motion of the satellites etc.

What is wrong? Newton theory of gravity is not Lorentz-invariant. Observations! The motion of the perihelion of Mercury is not described by this law.

Intermediate results Correct theory must meet a certain set of fundamental properties (axioms). General relativity is actually not the only possible theory of gravity (generalizing - field theory). The question arises whether it is possible to determine the form of the theory (i.e. the form of the equations describing its basic laws), based only on general principles, that is, completely disregarding the observations.

Kinetic energy Is it possible:

Kinetic energy Is it possible: ?

Kinetic energy Is it possible:

Kinetic energy Is it possible:

An example: square latice

Intermediate results Correct theory must meet a certain set of fundamental properties (axioms). One of them - Lorentz invariance. General relativity is actually not the only possible theory of gravity (generalizing - field theory). In the limit of weak fields and low speeds we have to return to the old theory. A big role to play invariants.

Important conclusions General principles (symmetry, Lorentz invariance) can help limit the theory, but, in general, do not define it through. When extending the theory necessary to introduce dimensional constants (mass M, velocity c ), the values of which can only be determined from observations. In the limiting case (in the example above - at non-relativistic velocities v << c ) theory should be returned to well-known. One possible generalization - the transition from scalars (numbers) to tensors (tables).

An example – standard model

An example – (non)standard model

Appearing of tensors Quadratic forms kinetic energy metric Linear dependence Hooke's law Ohm’s law

Homogeneous media Hall current

Invariants of matrices '‘Square'' of symmetric matrix The sum of the diagonal elements - the so-called 'Trace' (in German 'Spur')

Problem Show that a square matrix is independent of the angle 

Metric

Cartesian Cylindrical Spherical

Metric Arbitrary

Metric En example – oblique grid Problem

And what with the invariants? For orthogonal coordinate (i.e. for the diagonal matrices)