Einstein was no stranger to mathematical challenges. He struggled to define
energy in a way that acknowledged both the law of energy conservation and
covariance, which is general relativity's fundamental feature where physical
laws are the same for all observers.

A research team at Kyoto University's Yukawa Institute for Theoretical
Physics has now proposed a novel approach to this longstanding problem by
defining energy to incorporate the concept of entropy. Although a great deal
of effort has gone into reconciling the elegance of general relativity with
quantum mechanics, team member Shuichi Yokoyama says, "The solution is
shockingly intuitive."

Einstein's field equations describe how matter and energy shape spacetime
and how in turn the structure of spacetime moves matter and energy. Solving
this set of equations, however, is notoriously difficult, such as with
pinning down the behavior of a charge associated with an energy-momentum
tensor, the troublesome factor that describes mass and energy.

The research team has observed that the conservation of charge resembles
entropy, which can be described as a measure of the number of different ways
of arranging parts of a system.

And there's the rub: Conserved entropy defies this standard definition.

The existence of this conserved quantity contradicts a principle in basic
physics known as Noether's theorem, in which conservation of any quantity
generally arises because of some kind of symmetry in a system.

Surprised that other researchers have not already applied this new
definition of the energy-momentum tensor, another team member, Shinya Aoki,
adds that he is "also intrigued that in general curved spacetime, a
conserved quantity can be defined even without symmetry."

In fact, the team has also applied this novel approach to observe a variety
of cosmic phenomena, such as the expansion of the universe and black holes.
While the calculations correspond well with the currently accepted behavior
of entropy for a Schwarzschild black hole, the equations show that entropy
density is concentrated at the singularity in the center of the black hole,
a region where spacetime becomes poorly defined.

The authors hope that their research will spur deeper discussion among many
scientists not only in gravity theory but also in basic physics.

## Reference:

Sinya Aoki et al, Charge conservation, entropy current and gravitation,
International Journal of Modern Physics A (2021).
DOI: 10.1142/S0217751X21502018

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