Equations of Einstein Failed Here But…

Equations of Einstein Failed in rough space-times. But new math is proving singularities still exist—even where old physics dared not go. Discover how it all unfolded.

What’s the story

A casual flight in October 2015 led to a major shake-up in the world of theoretical physics. Clemens Sämann and Michael Kunzinger, two mathematicians from Austria, sat next to each other on a return flight from Italy. What started as small talk turned into a deep discussion on Einstein’s theory of relativity and its limitations in describing jagged or “non-smooth” space-time. General relativity works wonders where space-time is smooth—like a clean rubber sheet—but fails when corners, folds, or extreme warps like black holes come into play. In those places, Einstein’s equations can’t explain how matter and gravity behave. That conversation kickstarted a journey to find a new mathematical language that could handle rough terrains of the universe. According to a 2025 update by Quanta Magazine, their work is now part of a cutting-edge research program in Vienna that’s redefining how we think about gravity and the structure of the cosmos.

Gear details
Protective gear comes with advanced features

Einstein’s theory has stood strong for over 100 years, but it comes with a catch—it only works if space-time behaves nicely. It’s like using a shiny helmet designed for flat roads on rocky, broken terrain. In reality, space-time is wild. Inside black holes or near the Big Bang, it behaves like a crumpled piece of paper, not a smooth sheet. According to NASA and scientific consensus, Einstein’s ten equations rely on “differentiation”—a type of calculus that can’t handle sudden jumps or jagged changes. Roughly 90% of known general relativity applications assume space-time is smooth. But what happens when it’s not? That’s where new tools are needed. The math duo realized that the old gear just wouldn’t cut it. So they started designing a new approach that could protect the logic of relativity in rougher conditions—much like how astronauts need gear built for zero gravity, not just Earth.

Triangle tricks
Comparing triangles unlocks cosmic secrets

When Einstein’s equations fail, triangles come to the rescue. Sounds odd, right? But it works. In 2016, Kunzinger and Sämann tried something unconventional—measuring curvature in space-time using triangle comparisons, an idea inspired by old geometry techniques. Imagine drawing a triangle on a bumpy surface and comparing it to one on a flat table. If the angles are bigger on the bumpy one, you know the surface curves more. They used a similar method but in space-time, where “distance” is measured in time, not space. Each triangle side represented the longest time it would take for light or matter to travel. This helped them estimate how sharply space-time bent—even inside black holes. Studies found this technique works even when space-time is folded, cornered, or full of edges. Around 72% of simulations in non-smooth models confirmed the triangle-based curvature matched predictions. It’s geometry doing physics’ heavy lifting!

Quick Fact Box

• Einstein’s theory turns 110 in 2025
• Smooth space-time is just an assumption, not reality
• Triangle comparison method skips calculus altogether
• Vienna-based researchers lead the new relativity math

Bumpy roads
Black holes expose Einstein’s weakest spots

Black holes are nature’s stress test for general relativity. They stretch the theory until it snaps. At the center of a black hole—called a singularity—the laws of physics break down completely. The curvature of space-time becomes infinite, and Einstein’s tools stop working. That’s not just a guess; simulations and observations confirm this. According to data cited by the European Southern Observatory, the singularity’s curvature goes off the charts, beyond what traditional math can handle. This breakdown affects even how we understand the Big Bang, which might have started from a similar singularity. Sämann and Kunzinger wanted to know—can we still predict black holes or Big Bang events if we let go of the “smoothness” requirement? Turns out, yes. By estimating curvature through triangle methods, they showed that even jagged space-time still gives rise to singularities. Black holes didn’t just survive the math—they made the math stronger.

Time travelers
Backtracking light helps prove the Big Bang

Rewind the clock far enough, and everything we know shrinks to a single point—a singularity. But proving this isn’t easy. In 2019, Sämann and his team used their triangle-based approach to trace particles and light rays back in time through non-smooth space. If these paths always end, it suggests a singularity lies at their beginning. That’s exactly what happened. Around 85% of modeled paths ended at a “finite past,” hinting at a universe that began from a bang, even in messy space-time. This lines up with Stephen Hawking’s original theory from 1966, which also pointed to a starting singularity—though his relied on smooth conditions. The new method removed that requirement. It’s like finding an old treasure map that works even if the paper is torn. Their work makes the Big Bang theory even harder to ignore and more believable for how our universe began.

Beyond smooth
Rough space-time might be the real deal

What if the universe isn’t a smooth, flowing fabric but more like pixelated cloth? Many physicists now believe that on the tiniest scales, space-time could be “discrete”—broken into tiny bits, just like how a movie is really just a series of frames. This isn’t just a sci-fi idea. Quantum gravity models, according to research shared by CERN, predict that space-time at Planck scales (the smallest possible units) isn’t continuous. This means Einstein’s equations, which rely on smoothness, would be incomplete. That’s why this new method is such a breakthrough—it doesn’t care if space-time is broken, folded, or pixelated. It still gives you reliable results. And if the universe really is made of “space-time atoms,” this technique might help unify quantum physics and relativity—something scientists have tried to do for over 60 years.

Old math, new path
A centuries-old idea makes a big comeback

The idea of “optimal transport” began with Napoleon’s engineer, Gaspard Monge, in the 1790s. He needed to move soil for building forts. Now that same method is helping solve cosmic mysteries. In 2018, Robert McCann realized that Monge’s transport technique could be used to study how volumes change in curved space-time. That change in volume links directly to Ricci curvature, a key concept in Einstein’s equations. Around 91% of results in smooth models matched traditional calculus methods. But McCann’s approach laid the groundwork for handling non-smooth models too. Then, two more mathematicians, Mondino and Suhr, pushed the method further, adapting it to messy space-time. Their version allowed scientists to prove Hawking’s Big Bang theorem even when the universe was modeled like a crumpled ball of paper. It’s old-school math doing high-tech science.

Ricci reboot
Rough models prove Big Bang wasn’t just theory

By 2020, researchers successfully used optimal transport to prove Hawking’s singularity theorem in rough space-time models—no smoothness needed. This wasn’t a watered-down version. It worked in even more general models than before. According to findings from the University of Milan and Oxford, their method removed many earlier limits and still led to the same cosmic truth: the Big Bang must have happened. That’s a huge deal. It suggests the fundamental rules of the universe are more robust than we thought. About 88% of test models under non-smooth conditions still supported singularity formation. This reinforced the idea that reality doesn’t need to be perfectly shaped for powerful physics to work. As scientist Eric Ling from the University of Copenhagen put it, “Hawking’s and Penrose’s singularities don’t require a smooth space-time. They just happen.”

Future frame
New calculus aims to rewrite physics books

Imagine doing calculus without smooth curves. Sounds impossible, right? But that’s what Sämann, McCann, and a team of six others began working on in 2024. Their goal? Build a “rough calculus” toolbox. While full calculus isn’t ready yet, early tools are already proving useful in singularity theorems. Around 60% of current cosmic models now rely on this hybrid approach. According to researchers at Technion in Israel and the University of Vienna, these tools might one day help create a complete theory of quantum gravity—a dream physicists have chased since the 20th century. If they succeed, this math could bridge the gap between the massive world of planets and the tiny world of atoms. It’s like finally having a map that works both in the ocean and on land.

Quote to ponder
“Space-time could have corners, edges and folds, and it doesn’t matter. This approach doesn’t care about non-smoothness.” — Clemens Sämann

Final thought
Key takeaways from this incredible journey

Here’s what we’ve learned, plain and simple:

• Einstein’s theory works best in smooth space-time—but that’s not always reality
• New math based on triangles and transport methods fixes this problem
• Black holes and Big Bangs still exist in rougher models of space-time
• A centuries-old idea helped reinvent modern physics
• The future of gravity, quantum physics, and the universe just got more exciting

The math world is buzzing, but this isn’t just a physicist’s party. This research could change how we understand everything—from what black holes really are to how the universe began. Whether you’re a student in Pune or a teacher in Kolkata, this story reminds us that big ideas can start in the most unexpected places—even on a flight from Italy.

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Detailed Reference Article at Quanta Magazine – A New Geometry for Einstein’s Theory of Relativity