*06 May 2021*

# Einstein's General Relativity and Kerr Black Holes

Einstein’s theory of general relativity (GR) has stood strong for more than 100 years, but technology has given astronomers the chance to probe the limits of GR even further. The latest series of tests were done with data from a telescope that spans the entire globe and instruments so sensitive they can detect colliding black holes billions of light-years away…

Gravity is one of the four fundamental forces at work in the Universe. It’s also arguably the one with the highest-profile courtesy of a falling apple that hit Isaac Newton on the head and a rather well-known scientist named Albert Einstein. In an effort to understand exactly how it works, astronomers regularly test their theories in the most extreme environments known, and recent research has led them to conclude that we have a pretty decent idea of what is going on.

A collaboration of astronomers, including from the Arc Centre of Excellence for Gravitational Wave Discovery (OzGrav), has used observations of a supermassive black hole and gravitational waves to confirm some of Einstein’s mind-bending ideas on how gravity works. Specifically, they tested whether the shadow of a supermassive black hole and the inspiral phase of coalescing black holes were consistent with the Kerr metric.

More on the research later, but first some background on what all of that means.

Our best description of the Universe, what it looks like and how it behaves, is Einstein’s general relativity. Einstein developed his theories after it became apparent that the Newtonian idea that observers always have an equivalent view of the world wasn’t correct. It turns out that your description of nature depends on how you are moving.

Now, this is not something that we notice in our everyday lives. You’re not likely to observe the world behaving strangely as you are walking to your local café to grab a coffee. But when things are traveling at nearly the speed of light, things get very odd, very quickly.

Time progresses slowly. Distances shrink. And this is not conjecture; it has all been confirmed through many observations of our Universe by many scientists over the last 100 years.

At the heart of all of this is the idea that what we perceive as gravity is just a consequence of our motion through spacetime. The larger the curvature of spacetime, the stronger the gravity. And it is mass that causes spacetime to curve.

So, if you want to test these theories, you need somewhere with extreme amounts of matter packed into a relatively small space. Luckily, we know just where to look.

## The Kerr Metric

As you would imagine, the theory of general relativity is built on a lot of complicated mathematics. Einstein developed 10 field equations that describe gravity as being the result of spacetime having curvature by virtue of the mass and energy within it. Solving the field equations tells us how spacetime behaves under certain conditions, and those solutions have always been in close agreement with what we actually observe.

As an aside, though Einstein’s field equations have been fantastically helpful to us in understanding the Universe, when we move from the realm of the very large to the very small, they are incomplete. They are intrinsically incompatible with quantum theory in that they require energy and momentum to be defined precisely at every point in spacetime, which contradicts the uncertainty principle for quantum states. But that discussion is for another day…

One of the solutions to the Einstein field equations is the Kerr metric. A black hole is defined by just three properties: its mass, its spin, and its electric charge. The Kerr metric describes the geometry of an empty region of space surrounding a black hole that is uncharged but spinning. Exactly the kind of black holes that we expect to find scattered throughout the Universe.

The Kerr metric makes some really interesting predictions about the spacetime around a black hole that are a consequence of them having angular momentum. For example, rather than having a single event horizon, Kerr black holes have two, and instead of being neatly spherical, they are shaped like squashed spheres.

Even stranger though is that the singularity at the centre of a Kerr black hole is not a point, but a 1-dimensional ring. If the black hole was spinning fast enough, you would even be able to see it.

And guess what. When astronomers took the first-ever optical image of a black hole, at the centre of galaxy M87, using the Event Horizon Telescope (EHT), they found it to be spinning at 94% of its maximum speed with a 1-dimensional singularity larger than the orbit of Pluto and consistent with many of the predictions of the Kerr metric.

## Testing General Relativity

Indeed, any properties that are not compatible with a Kerr black hole could be seen as breaking general relativity. So the astronomers in this study took the data on the black hole and pushed it even further, calculating the size of the shadow cast by the monster and comparing that to theoretical values.

But they didn’t stop there. They also took data from two recent gravitational wave events detected by LIGO/Virgo, each the result of black hole collisions, and checked whether the pitch and intensity of the waves precisely matched predictions from general relativity. Inspiral gravitational waves are generated during the final moments before mergers of massive objects like black holes and neutron stars, and quickly increase in pitch and intensity as the objects get closer.

OzGrav researcher Ethan Payne, a co-author of the study, explained that the two measurements provided similar, consistent constraints. “Different sizes of black holes may help break the complementary behaviour seen here between EHT and LIGO/Virgo observations,” said Payne.

So now for the results of the research. How did general relativity hold up in two of the most extreme environments in the Universe? Very well, in fact.

Combining their measurements of the shadow of the black hole in M87 with gravitational wave events GW170608 and GW190924, the team found absolutely no evidence for any deviations from general relativity.

“This study lays the groundwork for future measurements of deviations from the Kerr metric,” said Payne.

Relativity is one of the most precisely tested theories in all of science. To date, it has passed every single test that scientists have thrown at it, proving over and over again that Einstein’s revolutionary ideas about gravity were right.

But somewhere in our future, chances are that the next Einstein is going to come along and poke the tiniest of holes in general relativity, just as Einstein did to Newton’s theory of gravity. And that will lead us to an even better understanding of the cosmos and of the nature of our own existence.

The paper appears on the arXiv preprint server...