New gravity theory may outdistance Einstein -- Part 2
by MIKE MARTIN, UPI Science Correspondent
STORRS, Connecticut (UPI) -- Philip Mannheim's theory of conformal gravity
originated with the renowned German mathematician Hermann Weyl, who unsuccessfully
used the concept to unify gravity and electromagnetism, two of nature's forces
that share remarkably similar properties.
Conformal gravity takes its name from a universal symmetry that involves both lengths and angles.
Consider a small square composed of 4 lines that form 4 angles. Changing
the length of the lines does not change the size or sense of the angles -- they
remain at 90 degrees in the same 4 corners. The same can be said of a square
formed on a globe that curves the angles -- changing, the size of the square
will not affect the angles. Both straight and curved squares exhibit conformal
symmetry, otherwise known as scale invariance: changing the scale -- or size
-- of the squares leaves their corner angles and their orientation unchanged
and symmetric.
Mannheim uses conformal, or scale invariance because he hopes it will better
describe how angles -- and therefore gravity -- behave as the universe "scales
up" -- that is, as distances increase dramatically in all directions. Angles
describe curves and curves, as Einstein proved, describe gravity.
Einstein took a similar approach but he based his work on the fundamental
invariance of length alone. Einstein knew that length is the same no matter
where it is measured. From this basic idea came the heart of general relativity
-- how length, width, height, and time, as 4 combined dimensions called "space-time"--
curve around massive objects regardless of where those objects are located.
"The geometrical structure of conformal gravity is actually based on
both lengths and angles," Mannheim told United Press International. "Thus
it contains all of the geometry of Einstein's gravity and conformal invariance
as well."
The idea of space-time curvature is fundamental to both conformal gravity
and general relativity. But the so-called "field equations" offered
by the theories that describe how space and time curve in the presence of mass
are different. In his field equations, Einstein used a mathematical object called
the "Ricci tensor" to describe curvature, named for the 19th century
Italian mathematician Gregorio Ricci.
Mannheim, on the other hand, uses a more complicated mathematical object
called the "Weyl tensor" named for Hermann Weyl, who died the same
year as Einstein -- 1955. Weyl was one of the first mathematicians to describe
conformal, or scale invariance, and the tensor that bears his name describes
how space and time curve under changes of scale.
The solutions of Mannheim's field equations that describe the behavior of
gravity involve extra terms that do not appear in Einstein's equations. Where
Einstein's solutions describe gravity with the term 2Gm/r, Mannheim's solutions
describe gravity with the term 2Gm/r + ar, where r is the distance from the
source of the gravitational field, G is Newton's gravitational constant, a is
a variable term, and m is the mass of a gravitating object.
As the universe scales up and the distance from the source of the gravitational
field -- Earth, perhaps -- increases, Mannheim's gravitational forces can grow
stronger and even become repulsive, while the reverse is true in Einstein's
case.
Changing the scale of the universe -- increasing distances in all directions
-- led Mannheim to adopt equations that he believes reflect the change better
than Einstein's equations.
As University of Texas astrophysicist Brad Schaefer points out, Mannheim is
not seeking to overthrow Einstein but to improve on Einstein's theory far from
the Earth and our solar system.
"Part of the allure is that Mannheim has a reasonable chance of improving
on the legendary Einstein," Schaefer told UPI from Austin. "An analogy
would be if a local guy started to make great improvements on the music of Mozart
and Beethoven."
Schaefer notes, however, that improvements alone have not led him to praise
Mannheim's work.
"Mannheim's gravity opens up many new ways of looking at the overall structure of the universe, and all cosmology will have to be redone if he succeeds, likely with both large and small changes," Schaefer said.