Two physicists derived Einstein’s equations of gravity from a measure of quantum information, a new clue in the search for quantum gravity.
If you open any physics textbook, it will tell you that gravity is one of the four fundamental forces of nature. However, two scientists have just shown that Einstein’s math for gravity can actually be explained by something that isn’t a force in the first place.
They start with a measure of how easily they can tell apart a patch of space holding matter and a patch of empty space.
If the argument holds, the curving of space and time isn’t a starting point. It is a consequence of how quantum information is arranged.
Philipp Dorau and Albert Much of Leipzig University in Germany published the work in March. It is a mathematical argument, not an experiment, and it rests on simplifying assumptions the authors spell out.
“It is a little like plucking a guitar string: matter is the pluck, spacetime is the string, and the note it produces is Einstein’s equations,” said Much, an assistant professor of quantum field theory and general relativity and a co-author of the study.
Jacobson’s idea, rewritten
In 1995, the physicist Ted Jacobson showed that Einstein’s equations could be derived from the physics of heat and entropy.
The laws describing a gas emerge from the motion of its molecules. Gravity, he suggested, might emerge from something deeper in the same way.
Jacobson’s version used classical thermodynamics, the science of heat flowing between warm and cold things. Dorau and Much wanted the same result using only quantum field theory, the mathematics that describes matter at the smallest scales.
For that they needed a different kind of entropy. The usual quantum measure of disorder gives infinite answers for a sharply defined region of space, because entanglement between neighboring points never stops at small scales.
Matter versus empty space
The quantity they used instead is called relative entropy. It measures how well two quantum states can be distinguished from each other, and it stays finite.
Here is one way to picture it. Imagine the surface of the sea, which is never perfectly still. Its constant rippling is like the quantum vacuum, the state physicists call empty space.
Now imagine a wave passing across that surface. Relative entropy tells you how well you can distinguish the wave from the rippling background.
The two physicists worked out this quantity at a horizon. Any small patch of curved space contains one, in the form seen by an observer who keeps accelerating.
“We show that this quantity, evaluated at a horizon – a boundary in spacetime beyond which no information can be received – equals the energy flowing across that horizon,” Much said.
From horizon to equations
The last step borrows a rule from black holes. In the 1970s, Jacob Bekenstein and Stephen Hawking argued that the entropy of a black hole grows with the area of its horizon rather than its volume.
Dorau and Much assume that same entropy-area rule holds for their local horizons. Relative entropy equals the energy crossing the horizon, that energy sets how much the horizon’s area changes, and Einstein’s equations follow with no extra ingredients.
With the black hole value for that rule, the usual constant in front of the equations comes out right. Furthermore, second free constant appears, which the authors identify with the cosmological constant.
However, the authors are careful about what they have shown. “Most of the ingredients for our work have been known for decades,” said Dorau, a theoretical physicist at Leipzig’s Institute for Theoretical Physics.
“It was recent progress in understanding relative entropy that gave us the missing link between quantum information, thermodynamics, and gravity.”
What the argument leaves out
The derivation treats matter in its simplest form, as the smallest possible disturbance of empty space. It works only to a first approximation.
The paper says a fully rigorous treatment would have to track smaller corrections, which the authors expect to be technically demanding.
It also leans on the equivalence principle, Einstein’s rule that any small enough region of curved space behaves like flat space. And the entropy-area relation at the heart of the final step is assumed, not derived.
What comes out at the end is the semiclassical form of Einstein’s equations, in which spacetime stays classical while matter is quantum.
Getting closer to quantum gravity
That isn’t yet a theory of quantum gravity, the long-sought merger of Einstein’s gravity with quantum theory. Furthermore, whether the same reasoning survives when matter takes more complicated forms is still an open question.
The two physicists plan to try. They want to extend the argument to other kinds of quantum matter, to more general quantum states, and to light-like surfaces, which are central to black hole physics.
“We are currently trying to generalize this to full quantum gravity, and it seems to me that this connection that we have identified will serve as a testbed for the various theories of quantum gravity,” Much said.
The full study was published in the journal Physical Review Letters.
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