Where did we start?Where did we start?

A conversation with Matti A conversation with Matti PitkanenPitkanen

Luis Eduardo Luna

Energy problem of general relativity

Two great theories of Einstein

1. Special Relativity

Symmetries of space-time give rise to conservation laws. Energy,

momentum, angular momentum are conserved.

Problem: No gravitation.

2. General Relativity

Curvature of space-time gives rise to gravitational ”force”

Problem: Because space-time is curved, the symmetries of empty space-time are lost. No conservation laws.

Why this problem was not taken seriously?

Gravitational interaction is extremely weak as compared to electromagnetism. So that one can say that the conceptual problem exists but does not have catastrophic effects as long as gravitational forces are weak and can be treated as small perturbations of empty space-time with symmetries.But can neglect so deep a conceptual

problem just because gravitational force is weak under ordinary circumstances? What about black holes?

What if we take this problem seriously? Can we imagine any solution to it?

A possible solution to the problem

Assume that space-time is 4-dimensional surface in some higher dimensional space-time. Assume that this higher-dimensional space-time is obtained from empty 4-dimensional space-time by replacing its points with some very small internal space.

This higher-dimensional spacetime has symmetries of empty spacetime so that the resulting physical theory has also these symmetries and the notion of energy is well-defined as in special relativity.

Gravitation can be described as curvature of space-time surface so that the theory describes also gravitation (see the illutration).

Replacing points of 1-dimensional line with small circles makes it a thin 2-dimensional cylinder

The 2-dimensional cylinder has symmetries of both line (translations) and circle (rotations)

Toy model with 1-dimensional spacetime in 2-dimensional space-time.

Realistic situation: replace 1-dimensional line with 4-dimensional surface and cylinder with D > 4-dimensional space-time

The physics for one-dimensional space-time (s) living in 2-dimensional cylinder space-time must have symmetries of cyclinder. Translations in the direction of axis of cylinder (time) imply conserved energy. Special relativity!

One-dimensional lines at cylinder can be also curved so that you have an analog of gravitation. General Relativity!

You have fusion of special and general relativities!

One-dimensional curved space-time on cylinder

What this means for 1-dimensional spacetimes living on cylinder space-time?

CONCLUSION:

You get a fusion of successful aspects of special and general relativities and get rid of their problems!

Additional bonus: the symmetries of the small internal space explain the other symmetries and conservatin laws of particle physics (electromagnetic charge,...)