Mathematical Formulation of the Yilmaz Theory of Gravity

where the lefthand side denotes the Einstein-Hilbert curvature tensor and the righthand side is the stress-energy tensor of the source fields (matter, electromagnetical...).

The new theory of Hüseyin Yilmaz can still be interpreted geometrically. What differs to general relativity is that the stress-energy-momentum of the gravitational field itself is assumed to be a source of curvature. Thus, the conventional theory of gravity has to be modified by adding the stress-energy tensor of the gravitational field to the righthand side of the Einstein equations [1]:

C. O. Alley states that he "believes that the explicit inclusion of the gravitational field stress-energy tensor [...] is as important for our understanding of physics as Maxwell's addition of the displacement current in his equations for the electromagnetical field" [2].

If one starts from the space-time metric, then the gravitational potential is defined (up to integration constants) according to

with

However, the geometrical interpretation of the new theory is not the essential one. Remarkably, the new theory is a gauge theory very similar to electromagnetism (see below). The Lagrangian of the free field (no other sources) is given by

with . The energy-momentum tensor of the gravitational field then follows by variation of the Lagrangian as

The inverted relation between the gravitational field and the space-time metric is formally exponential

The greek symbol eta represents the matrix of the usual metric of special relativity (Minkowski metric).

Differences to General Relativity

• There is local conservation of energy-momentum, as a definite stress-energy tensor can be assigned to the gravitational field.
• In the solutions of the field equations there appear no space-time singularities and there are no "event-horizons", in contrast to general relativity. Thus, Yilmaz's theory of gravity does not predict the existance of black holes. It is worth noting that Albert Einstein himself could never believe in the reality of black holes - he considered them to represent rather a mathematical anomaly than real objects.

• The new theory is compatible with all experimental data (what it is essential, of course).
• The strong equivalence principle, which means the equivalence of inertial mass, active gravitational mass and passive gravitational mass, is a direct consequence of a multiplicative gauge invariance. The equivalence principle is not deducible in Einstein's theory, where it has to be included as an additional assumption.
• There exist closed solutions for gravity waves for arbitrary field strengths. The wave equation (see below) is exactly satiesfied, and not only in the weak field limit as in general relativity.
• Remarkably, H. Yilmaz also claims to have found a way to quantize the theory. Usually, gauge theories with non-polynomial Lagrangians (as is Yilmaz theory) are considered to be non-quantizable, because they are known to be non-renormizable. Surprisingly, H. Yilmaz found a way to avoid this obstacle. By applying some simple physical principles, he claims to have discovered a quantization method which renders the usual renormalization procedure superfluous [4,5].

Theories of Gravity	Albert Einsteins General Relativity	Hüseyin Yilmaz Theory of Gravity
Type of theory	no gauge theory	gauge theory
Local conservation of enery-momentum?	No	Yes
Definite stress-energy tensor exists?	No	Yes
Exact closed gravity wave solutions for arbitrary field strengths?	No	Yes
True Lagrangian exists?	No	Yes
Interaction can be deduced from gauge principle?	No	Yes
Strong principle of equivalence?	Not deducible	Holds
Theory is quantizable?	No(?)	Yes(?)
Can be interpreted geometrically (as curvature theory)?	Yes	Yes
Solutions with space-time singularities ("black holes")?	Yes	No
Compatible with all observations?	Yes(?)	Yes(?)

Correspondence to Maxwell's Theory of Electromagnetism

	Electromagnetism (Maxwell's theory)	Yilmaz Theory of Gravity
Field equations (Lorentz gauge)
Gauge transformations
Gauge-covariant derivatives

denotes the d'Alembert operator. In the Lorentz gauge the potentials of the new theory thus satisfy a standard wave equation.

Outlook

References