The first-ever detection of gravitational waves was made by LIGO in 2015 and since then researchers have been trying to understand the physics of the black-hole and neutron-star mergers that create the waves. However, the physics is very complicated and is defined by Albert Einstein’s general theory of relativity.

Now Jiaxi Wu, Siddharth Boyeneni and Elias Most at the California Institute of Technology (Caltech) have addressed this challenge by developing a new formulation of general relativity that is inspired by the equations that describe electromagnetic interactions. They show that general relativity behaves in the same way as the gravitational inverse square law described by Isaac Newton more than 300 years ago. “This is a very non-trivial insight,” says Most.

One of the fascinations of black holes is the extreme physics they invoke. These astronomical objects  pack so much mass into so little space that not even light can escape their gravitational pull. Black holes (and neutron stars) can exist in binary systems in which the objects orbit each other. These pairs eventually merge to create single black holes in events that create detectable gravitational waves. The study of these waves provides an important testbed for gravitational physics. However, the mathematics of general relativity that describe these mergers is very complicated.

Inverse square law

According to Newtonian physics, the gravitational attraction between two masses is proportional to the inverse of the square of the distance between them – the inverse square law. However, as Most points out, “Unless in special cases, general relativity was not thought to act in the same way.”

Over the past decade, gravitational-wave researchers have taken various approaches including post-Newtonian theory and effective one-body approaches to better understand the physics of black-hole mergers. One important challenge is how to model parameters such as orbital eccentricity and precession in black hole systems and how best to understand “ringdown”. The latter is the process whereby a black hole formed by a merger emits gravitational waves as it relaxes into a stable state.

The trio’s recasting of the equations of general relativity was inspired by the Maxwell equations that describe how electric and magnetic fields leapfrog each other through space. According to these equations, the forces between electric charges diminish according to the same inverse square law as Newton’s gravitational attraction.

Early reformulations

The original reformulations of “gravitoelectromagnetism” date back to the 90s. Most explains how among those who did this early work was his Caltech colleague and LIGO Nobel laureate Kip Thorne, who exploited a special mathematical structure of the curvature of space–time.

“This structure mathematically looks like the equations governing light and the attraction of electric charges, but the physics is quite different,” Most tells Physics World. The gravito-electric field thus derived describes how an object might squish under the forces of gravity. “Mathematically this means that the previous gravito-electric field falls off with inverse distance cubed, which is unlike the inverse distance square law of Newtonian gravity or electrostatic attraction,” adds Most.

Most’s own work follows on from previous studies of the potential radio emission from the interaction of magnetic fields during the collision of neutron stars and black holes from which it seemed reasonable to then “think about whether some of these insights naturally carry over to Einstein’s theory of gravity”. The trio began with different formulations of general relativity and electromagnetism with the aim of deriving gravitational analogues for the electric and magnetic fields that behave more closely to classical theories of electromagnetism. They then demonstrated how their formulation might describe the behaviour of a non-rotating Schwarzschild black hole, as well as a black hole binary.

Not so different

“Our work says that actually general relativity is not so different from Newtonian gravity (or better, electric forces) when expressed in the right way,” explains Most. The actual behaviour predicted is the same in both formulations but the trio’s reformulation reveals how general relativity and Newtonian physics are more similar than they are generally considered to be. “The main new thing is then what does it mean to ‘observe’ gravity, and what does it mean to measure distances relative to how you ‘observe’.”

Alexander Phillipov is a black-hole expert at the University of Maryland in the US and was not directly involved with Most’s research. He describes the research as “very nice”, adding that while the analogy between gravity and electromagnetism has been extensively explored in the past, there is novelty in the interpretation of results from fully nonlinear general relativistic simulations in terms of effective electromagnetic fields. “It promises to provide valuable intuition for a broad class of problems involving compact object mergers.”

The research is described in Physical Review Letters.

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Quantum computers open the door to profound increases in computational power, but the quantum states they rely on are fragile. Topologically protected quantum states are more robust, but the most experimentally promising route to topological quantum computing limits the calculations these states can perform. Now, however, a team of mathematicians and physicists in the US has found a way around this barrier. By exploiting a previously neglected aspect of topological quantum field theory, the team showed that these states can be much more broadly useful for quantum computation than was previously believed.

The quantum bits (qubits) in topological quantum computers are based on particle-like knots, or vortices, in the sea of electrons washing through a material. In two-dimensional materials, the behaviour of these quasiparticles diverges from that of everyday bosons and fermions, earning them the name of anyons (from “any”). The advantage of anyon-based quantum computing is that the only thing that can change the state of anyons is moving them around in relation to each other – a process called “braiding” that alters their relative topology.

However, as team leader Aaron Lauda of the University of Southern California explains, not all anyons are up to the task. Certain anyons derived from mathematical symmetries appear to have a quantum dimension of zero, meaning that they cannot be manipulated in quantum computations. Traditionally, he says, “you just throw those things away”.

The problem is that in this so-called “semisimple” model, braiding the remaining anyons, which are known as Ising anyons, only lends itself to a limited range of computational logic gates. These gates are called Clifford gates, and they can be efficiently simulated by classical computers, which reduces their usefulness for truly ground-breaking quantum machines.

New mathematical tools for anyons

Lauda’s interest in this problem was piqued when he realized that there had been some progress in the mathematical tools that apply to anyons. Notably, in 2011, Nathan Geer at Utah State University and Jonathan Kujawa at Oklahoma University in the US, together with Bertrand Patureau-Mirand at Université de Bretagne-Sud in France showed that what appear to be zero-dimensional objects in topological quantum field theory (TQFT) can actually be manipulated in ways that were not previously thought possible.

“What excites us is that these new TQFTs can be more powerful and possess properties not present in the traditional setting,” says Geer, who was not involved in the latest work.

As Lauda explains it, this new approach to TQFT led to “a different way to measure the contribution” of the anyons that the semisimple model leaves out – and surprisingly, the result wasn’t zero. Better still, he and his colleagues found that when certain types of discarded anyons – which they call “neglectons” because they were neglected in previous approaches – are added back into the model, Ising anyons can be braided around them in such a way as to allow any quantum computation.

The role of unitarity

Here, the catch was that including neglectons meant that the new model lacked a property known as unitarity. This is essential in the widely held probabilistic interpretation of quantum mechanics. “Most physicists start to get squeamish when you have, like, ‘non-unitarity’ or what we say, non positive definite [objects],” Lauda explains.

The team solved this problem with some ingenious workarounds created by Lauda’s PhD student, Filippo Iulianelli. Thanks to these workarounds, the team was able to confine the computational space to only those regions where anyon transformations work out as unitary.

Shawn Cui, who was not involved in this work, but whose research at Purdue University, US, centres around topological quantum field theory and quantum computation, describes the research by Lauda and colleagues as “a substantial theoretical advance with important implications for overcoming limitations of semisimple models”. However, he adds that realizing this progress in experimental terms “remains a long-term goal”.

For his part, Lauda points out that there are good precedents for particles being discovered after mathematical principles of symmetry were used to predict their existence. Murray Gell-Man’s prediction of the omega minus baryon in 1962 is, he says, a case in point. “One of the things I would say now is we already have systems where we’re seeing Ising anyons,” Lauda says. “We should be looking also for these neglectons in those settings.”

The research is published in Nature Communications.

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