## Using math!

Thanks to the two direct observations of gravitational waves, both of which likely originated from the merger of two black holes, scientists are pretty sure black holes exist and are more than just conjecture.

But what a black hole looks like is anyone’s guess because we have never been able to see inside one. Why? It’s the fault of their immense gravitational pull, gobbling up nearly every last bit of anything that mistakenly travels beyond the event horizon — where the escape velocity (or speed required to escape the gravitational pull of the black hole) is equal to the speed of light. Not even light can escape.

What this means is that none of humanity’s sophisticated devices can see what is going on inside these mysterious entities. Rather, scientists use mathematics. According to a new study by researchers of John Hopkins and Towson University, to figure out what’s inside a black hole, scientists need to focus on mathematical quantities known as invariants.

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Previously, physicists constructed their view of a black hole’s internal structure based on how certain mathematical coordinates fit together. However, there is a problem with this method. Depending on the coordinates chosen, and how they’re viewed based on your position as an observer, you will likely get very different results from someone who chose a different set of coordinates from another viewpoint.

Enter invariants, which have the same value for any coordinate system. "The truest way to depict the properties of a black hole is through quantities that are coordinate-invariant," explained the researchers in their paper, which has yet to be peer-reviewed.

According to the authors, there are 17 invariant quantities related to the curvature of spacetime that could be used to study black hole interiors, but because of certain mathematical relationships between them, just five are exactly independent.

"[O]ne needs five such quantities to fully characterise the curvature of spacetime inside all possible time-independent black holes," the team reported.

What did they see? Something awesome:

When they mapped the structure of a black hole using the invariants, they saw something awesome: "a landscape that is much more beautiful and complex than usually thought." Take a look.

^{One of the invariants plotted for several values of r (radius of the event horizon) in spherical coordinates. Photo credit: Richard Conn Henry, James Overduin, and Kielan Wilcomb}

This image looks substantially different to a standard image of a black hole, and its curvature is "far from constant."

Now that their paper is published online, other physicists can use the five invariants to also construct the interior of a hypothetical black hole. The results should lead to more insight into the nature and structure of black holes.

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