The Biggest Misconception in Physics

An astronaut in deep space throws a rock as hard as possible. According to Newton's first law, the rock should continue with constant velocity in a straight line.

In reality, the rock eventually slows down and stops, contradicting the expectation from classical physics.

Where does the rock's energy go? This question baffled physicists at the turn of the 20th century, including Albert Einstein.

Emmy Noether, an unpaid mathematician, proved Einstein wrong and created a paradigm that underlies all modern particle physics and explains conservation laws.

In 1915 at the University of Göttingen, Einstein delivered six lectures on his new theory of gravity, which would become general relativity.

Einstein struggled to show that total energy was conserved in his new theory. The question arose: is energy in the curvature of spacetime or in the stress-energy tensor?

Mathematician David Hilbert looked for energy conservation equations in Einstein's theory but found only the Bianchi identities, which showed energy conservation only in a completely empty universe.

Hilbert turned to his assistant Emmy Noether, recognizing she was the right person to solve this fundamental problem.

From an early age, Noether wanted to follow her father's footsteps as a mathematics professor. The Erlangen Academic Senate refused to admit her as an official student, claiming women's admission would 'overthrow all academic order.'

In 1903 at Göttingen, Noether learned about a new approach to geometry using symmetry, which would become central to her later work.

Over the next 12 years, Noether became a leading expert on symmetry and was only the second woman in Germany to earn a PhD in mathematics.

She used her expertise in symmetry to help Hilbert and Einstein solve their problem of energy conservation in general relativity.

An equilateral triangle has three axes of symmetry. When reflected about these axes, the triangle looks the same and remains unchanged.

Mathematical functions can have translation symmetry: if you shift a function up or down by any constant, its derivative (slope) remains unchanged regardless of the constant added.

Mathematicians generalize symmetry as any action you can take that leaves an object unchanged, including rotations, reflections, and abstract transformations.

Noether's deep knowledge of symmetry enabled her to connect it to conservation laws in physics, fundamentally changing how physicists understood the universe.

Einstein imagined a window cleaner falling from a building and realized that during free fall, the person would feel weightless. This led to the equivalence principle: acceleration and gravity are equivalent.

Einstein required that the laws of gravity have the same form in every frame of reference, a core principle called general covariance.

To satisfy general covariance, Einstein used tensors, special mathematical objects whose components change when coordinate systems change, but the tensor itself remains invariant.

Noether recognized that Einstein's proposed energy conservation equation contained a pseudotensor, which violates the principle of general covariance because it doesn't remain the same across different reference frames.

In an empty, static universe with translation symmetry, a thrown ball travels at constant speed. This symmetry directly leads to conservation of momentum.

If the universe is symmetric under rotations, a spinning object continues to rotate indefinitely. This symmetry corresponds to conservation of angular momentum.

If the laws of physics don't change over time, an experiment today gives the same results as tomorrow. This symmetry leads to conservation of energy.

Noether proved that anytime there is a continuous symmetry, there is a corresponding conservation law. This theorem reveals the deep origin of all conservation laws in physics.

Everything follows the path that minimizes a quantity called action, equivalent to the integral of the Lagrangian over time.

If an experiment gives the same result now as after a tiny time interval, the Lagrangian changes by dL/dt times that interval, but this change doesn't affect the equations of motion.

• Using the chain rule and Euler-Lagrange equations to expand dL/dt • Applying the product rule in reverse to simplify the expression • Finding that the time derivative of a specific quantity equals zero

The conserved quantity equals kinetic energy minus potential energy plus potential energy, which simplifies to total energy. Time translation symmetry is directly equivalent to energy conservation.

In the 1920s, astronomers discovered that all distant galaxies are moving away from us, and in the 1990s, precise measurements revealed that this expansion is accelerating.

Since the universe expands and changes dramatically over billions of years, it lacks time translation symmetry. The universe 13 billion years ago was very different from today.

• A photon of visible light from 380,000 years after the Big Bang has lost 99.9% of its energy by the time it reaches us • The rock thrown in space slows down as it travels through the expanding universe, losing energy that doesn't go anywhere

On human timescales of days or years, time translation symmetry approximately holds, so energy appears conserved. Only on timescales of millions of years does the universe's expansion become significant enough to break this symmetry.

In general relativity, you cannot shift the entire universe and have physics remain the same because spacetime curvature varies from point to point.

General covariance creates local symmetries: in any small region, you can change your frame of reference, and the laws of physics always look the same.

Noether's second theorem proved that local symmetries don't give proper conservation laws but instead give continuity equations that work only locally.

Energy is conserved in small patches of spacetime, but when linking patches together, spacetime curvature creates 'cracks' where energy can leak out, accounting for changes in the gravitational field.

Noether discovered that her continuity equation was exactly equivalent to the Bianchi identities that Hilbert had dismissed as useless.

While the Bianchi identities only showed energy conservation in an empty universe, Noether proved it was the best result possible in general relativity due to its local nature.

• Uncovered the source of all conservation laws through symmetry principles • Solved the energy conservation problem in general relativity that eluded both Hilbert and Einstein

These two theorems are considered among the most important theorems in 20th-century physics.

The University of Göttingen made Noether's position official, allowing her to teach, and in 1923 she became a professor with a small salary.

On January 30, 1933, Hitler became chancellor of Germany. The Nazis banned Jewish people from working at universities, and Noether was immediately suspended.

Despite her dismissal, Noether continued teaching mathematics in her home kitchen. When a former student in a Nazi stormtrooper uniform came to learn, she welcomed him and taught him without discrimination.

With help from other academics, Noether obtained a teaching position at Bryn Mawr, a women's college in America, where she taught until her death. Einstein wrote in her New York Times obituary that she was 'the most significant creative mathematical genius thus far produced since the higher education of women began.'

Physicists fundamentally changed their thinking about physics, beginning to analyze everything in terms of symmetries rather than forces.

• Discovered that charged particles like electrons have phase symmetries • Found that offset or gauge symmetry leads to conservation of electric charge

In the 1960s and 1970s, Noether's insights directly led to the discovery of fundamental particles like quarks and the Higgs boson.

• Explained where the forces of nature originate • Helped explain the origin of all mass in the universe • Brought us closer to a theory of everything than any other framework