# Toward a New Theory of Gravity (Solves Incompatibility with Quantum Mechanics)

General Relativity’s Major Problem

• The thought experiment at No Black Holes shows that general relativity (GR) contradicts itself about black holes.
• It shows that a self-consistent theory of gravity that postulates the equivalence principle must preclude black holes, in which case:
• The escape velocity in the theory must be always less than c, the speed of light, such that escape is always possible in principle.

Proposal for a New Theory

• The new theory of gravity postulates the equivalence principle. Then:
• Special relativity applies in any locally inertial frame.
• A new metric is given for Schwarzschild geometry only. That’s the geometry of empty spacetime around a spherically symmetric, uncharged, non-rotating center of gravitational attraction.
• The vast majority of tests of GR have been tests of the Schwarzschild metric. The new metric replaces it.
• The derivation of the new metric isn’t given here, but is available upon request. The scientific method doesn’t require derivations; the Einstein field equations weren’t originally derived, for example.
• The scientific method doesn’t require field equations for a theory of gravity. An equation that predicts observations is sufficient. That said, the Einstein field equations could be modified so they yield the new metric as their solution for Schwarzschild geometry, instead of the Schwarzschild metric.

New Metric for Schwarzschild Geometry

• The new metric for Schwarzschild geometry, for a spatial plane through the center of a spherically symmetric, uncharged, non-rotating center of gravitational attraction is (in geometric units):
• Timelike form:
$d\tau^2=\left(\displaystyle\frac{r}{r+2M}\right)dt^2-\displaystyle\frac{dr^2}{\left(\displaystyle\frac{r}{r+2M}\right)}-r^2d\phi^2$
• Spacelike form:
$d\sigma^2=-\left(\displaystyle\frac{r}{r+2M}\right)dt^2+\displaystyle\frac{dr^2}{\left(\displaystyle\frac{r}{r+2M}\right)}+r^2d\phi^2$
• Where:
• (“dee phi”) = Increment of an angle in a plane through the center of gravitational attraction.
• r = Reduced circumference, defined so that the circumference of a sphere at radius r is 2πr.
• dr = Increment of radial separation, where r is the reduced circumference.
• (“dee sigma”) = Increment of proper distance between two adjacent events.
• dt = Increment of time between two adjacent events as measured by a sufficiently faraway observer (technically, from infinity).
• dτ (“dee tau”) = Increment of wristwatch time (proper time) between two adjacent events.
• M = Mass of the center of gravitational attraction.
• In weak gravity the Schwarzschild metric approximates the new metric. I know of no experiment of Schwarzschild geometry for which the results of the metrics differ after rounding for significant digits. This includes the anomalous perihelion shift of Mercury (the new metric predicts 42.98 arcseconds per century, the same as the Schwarzschild metric does) and, in stronger gravity, the observations of the binary pulsar system PSR J0737-3039.
• Black holes aren’t predicted by the new metric (see below).
• An object may still appear black due to a very high (finite) gravitational redshift of its light.

New Equation for Escape Velocity

• The new equation for escape velocity, derivable from the new metric above, is (in geometric units):
• $v=\sqrt{\displaystyle\frac{2M}{r+2M}}$
• Where:
• v = velocity as a fraction of c.
• M and r are defined as above.
• Since v < c when r > 0, no body need collapse to create a black hole (as is the case in GR when the reduced circumference r of the surface of a body becomes <= 2M). Then:
• r = 0 can be a limiting case only; gravitational singularities needn’t exist. (In other words it can be that a center of gravitational attraction always has a positive volume.) Then:
• The new theory, unlike GR, isn’t incompatible with quantum mechanics due to predictions of black holes and their singularities. Then:
• Occam’s razor strongly favors the new metric over the Schwarzschild metric, even ignoring the latter’s incompatibility with the equivalence principle.