The Biggest Misconception in Physics

Noether's theorem reveals why energy fails to conserve in General Relativity: the expanding universe breaks time symmetry.

Core Insight

• Continuous symmetries in the laws of physics directly produce conservation laws.
• Time-translation symmetry yields energy conservation; its absence in an expanding cosmos means energy is not conserved (20:06-27:32).

Three Symmetries

• Spatial translation symmetry conserves linear momentum.
• Rotational symmetry conserves angular momentum.
• Time translation symmetry conserves energy (10:02-20:05).

General Relativity Problem

• Einstein's proposed energy equation violated general covariance.
• Noether demonstrated that local symmetries in GR do not produce global conservation laws (0:00-10:01).

Expanding Universe

• Photons lose energy over cosmic distances as wavelengths stretch.
• A rock thrown in deep space eventually stops because time symmetry does not hold at large scales.

00:00:00 What is symmetry?

Speaker: Derek

• 00:00:00 Imagine you are an astronaut out drifting in deep space when you throw a rock as hard as you can. What's gonna happen to that rock? You would think that it would continue with constant velocity in a straight line. That's just Newton's first law. But what actually happens is it eventually slows down and stops. So why does this happen? Where did all the rock's energy go? (pensive orchestral music)
• 00:00:25 At the turn of the 20th century, the problem of energy conservation baffled some of the greatest minds, including Albert Einstein. Einstein came up with a possible solution, but then a little-known unpaid mathematician named Emmy Noether proved he was wrong. And in doing so, she created a whole new paradigm for physics, one that underlies all of particle physics and explains why anything is conserved. (gentle inquisitive music)
• 00:00:55 It all started in 1915 at the University of Gottingen, where Einstein was giving six lectures on his new theory of gravity. What would become the general theory of relativity. The lectures were well received, but Einstein hadn't yet settled on the final form of the field equations. One problem he was facing was how to show that total energy was conserved in his new theory.
• 00:01:25 > "And this is the whole beginning of this story, right? Classically, they thought they had this understanding of what the energy of the gravitational field was. All of a sudden with these new equations, they go, 'Where is it? You know, is it in the curvature? You know, is it in the stress-energy tensor? Where's the term that we're looking for?'"
• 00:01:50 Einstein suggested that the principle of conservation of energy, long established as a bedrock of physics, might hold the key to working out the correct field equations. In the audience, legendary mathematician David Hilbert was intrigued. So he started to look for the energy conservation equations in Einstein's new theory, but the best he could find was a set of equations known as the Bianchi identities. They showed that energy was conserved, but only in a completely empty universe. So for one like ours filled with stuff, they seemed useless. Hilbert was stumped.

⭐ HIGH PRIORITY

• 00:02:35 Fortunately, he knew just the person for the job. His new assistant, Emmy Noether. (light thoughtful music)
• 00:02:55 From an early age, Noether had dreamed of following the footsteps of her father, a mathematics professor at the University of Erlangen. She got special permission to attend lectures at the university, but they refused to admit her as an official student. The Erlangen Academic Senate held that the admission of women "would overthrow all academic order." (lively orchestral music) So in 1903, she spent a semester at Gottingen instead. There she learned about a new way to approach geometry using symmetry. (gentle thoughtful music)
• 00:03:40 Symmetry is one of those ideas that's easy to recognize but harder to describe. - If I align a mirror on this triangle like so, then it looks the same as without the mirror, and that's because this is an axis of symmetry. The reflections about this axis leave the triangle unchanged. And the same thing happens if I put the mirror like this or if I orient the mirror like this. So this triangle has three axes of symmetry. - 00:04:10 Now, mathematicians generalize the idea of symmetry further to be any action you can take that leaves an object unchanged. So something else I could do is I could rotate this triangle by 120 degrees or by 240 degrees or by 360 degrees. Together, these six actions capture all the symmetries of the equilateral triangle.
• 00:04:40 <span>But you can also have more abstract symmetries, for example, with a mathematical function. If I shift this function up or down by some constant amount, call it a, then all of its y values will change. But if I differentiate that function, I get the slope, and that remains unchanged regardless of whatever constant is added. So there is a kind of translation symmetry. And unlike the symmetries of the triangle, this is a continuous symmetry, meaning you can shift it by any amount you like.</span>
• 00:05:20 Over the next 12 years, Noether became a leading expert on symmetry. She became only the second woman in Germany to earn a PhD in mathematics, and she used this expertise to help Hilbert and Einstein with their problem of energy conservation. The issue had bothered Einstein so much that he proposed a new conservation equation. It said that if we add together the energy of matter and the energy of the gravitational field, then that total remains constant. Its change over time and space is zero. But when Noether saw this, she was convinced Einstein had made a fundamental mistake because this equation disregards the foundational principle that general relativity is built on.
• 00:06:35 10 years earlier, in 1905, Einstein had introduced his special theory of relativity, and it was built on the idea that the laws of physics were independent of your frame of reference. But so far Einstein had only applied this principle to inertial frames of reference. Those are frames that move at a constant speed. - He began wondering what would it take to generalize that, to consider more general states of motion. After all, trains on the platform; he loved trains. Trains would speed up or slow down. People, you know, moving around the world don't only move at a single constant speed forever. - In 1907, he wrote, "Is it conceivable that the principle of relativity also applies to systems that are accelerated relative to each other?" This made him wonder; perhaps his principle could also be applied to accelerating and rotating frames, frames that move in any way in general. That's the "general" in general relativity. So Einstein got to work on this largely intellectual pursuit.
• 00:07:50 But to achieve this, Einstein needed to make sure the laws of gravity had the same form in every frame of reference. (train whooshing) This is the idea of general covariance, and it's one of the core tenets of general relativity. To satisfy it, Einstein knew he had to use special mathematical objects called tensors. A simple kind of tensor is a vector. You can write a vector as a set of components multiplied by their basis vectors. For example, this vector can be written as 3x hat plus 2y hat. But I can also write this using a different coordinate system. And with these new basis vectors, the original vector is now written as 2a plus 1b. So the components, the numbers in this list, changed, but the vector didn't. And that's because when the basis vectors change, the components adjust in a complementary way to keep the vector the same. The vector itself is independent of which coordinate system you use. And the same is true for tensors, only now instead of having just two components, a general tensor can have any number of them in the form of a matrix. And just like with vectors, you can change a tensor from one coordinate system to another, and the tensor stays the same. So that's why Einstein had to use them to build his new theory.
• 00:09:40 And that was exactly the problem that Noether found because when she looked at Einstein's proposed energy conservation equation, it contained a pseudotensor. And as the name implies, that isn't quite a tensor. When you try to transform it from one frame of reference to another, it doesn't remain the same quantity in different frames. The gravitational energy you might observe in one frame completely disappears in another.
• 00:10:15 Einstein, you know, he had some strange thoughts about this. I mean, people were trying to stamp conservation of energy into relativity by bending the rules of mathematics. - So, Noether knew that Einstein's proposed solution couldn't be the answer, and that made her think, "What if general covariance and energy conservation are simply incompatible? And if that's the case, then why?" General covariance says that laws of physics must stay the same when you change reference frames. So that is a kind of symmetry, exactly what Noether had spent her career studying. So she started thinking about the symmetries of the universe, beginning with the simplest possible case, an empty static universe. (light contemplative music)

04:25:00 Emmy Noether and Einstein

Speaker: Derek

• 04:25:00 Imagine you throw a ball in an empty, static universe. If you shift the whole universe in space, the physics stays the same. The result is that momentum is conserved due to spatial translation symmetry. If you rotate the universe, physics stays the same; angular momentum is conserved due to rotational symmetry. If you run a process today or tomorrow, physics is the same; time translation symmetry implies energy conservation. This is Noether’s First Theorem: every continuous symmetry of the action leads to a conservation law.
• 04:55:00 Noether’s insight was that energy conservation is not a separate axiom but a consequence of a deep symmetry: time translation symmetry. The simplest Lagrangian example is L = T − V, with kinetic energy T = 1/2 m v^2 and potential energy V. The mathematics shows that the time derivative of a particular combination is zero, which is the conserved energy. This is the core message: energy conservation follows from a symmetry of time.
• 05:40:00 But the universe we live in is not a static empty box. The cosmos is filled with matter, radiation, and gravity, and it is expanding. In the 1920s and beyond, astronomers observed galaxies receding, implying a cosmological time asymmetry. In other words, the universe is not symmetric under time translations on large scales. Noether’s Second Theorem deals with local (gauge) symmetries, showing that local symmetries lead to continuity equations rather than simple global conservation laws.

⭐ HIGH PRIORITY

• 06:50:00 In a curved spacetime, the gravity field itself contributes to energy and momentum in a way that is not neatly captured by a single tensor. This is where the Bianchi identities come into play. Hilbert found them first, but they only seemed to guarantee energy conservation in an empty universe. Noether showed that local symmetries—general covariance—imply a continuity equation that holds patch by patch, with energy exchange between matter and the gravitational field accounted for by curvature terms.

07:33:00 General Covariance

Speaker: Derek

• 07:33:00 General covariance means the laws of physics have the same form in every local frame of reference. To formalize this, you use tensors. A tensor is a geometric object that transforms in a specific way when you change coordinates, so its intrinsic meaning is independent of the coordinate system. A vector is the simplest tensor, but there are higher-rank tensors that generalize this idea. If you rotate or translate your coordinate axes, the components change, but the object itself remains the same. This is crucial for GR because gravity is encoded in the geometry of spacetime, not in a force in a fixed background.
• 08:40:00 The catch is that the energy of the gravitational field cannot be described by a proper tensor in GR. The gravitational energy is described by a pseudotensor, which does not transform as a tensor under general coordinate changes. This is why the total energy cannot be unambiguously defined in a curved spacetime and why energy conservation becomes tricky in GR.

11:59:00 The Principle of Least Action

Speaker: Derek

• 11:59:00 The action is the integral of the Lagrangian over time. The principle of least action states that the actual path taken by a system between two configurations is the one that minimizes (or extremizes) this action. Euler–Lagrange equations are the differential equations that result from applying this principle. Noether connected these equations to symmetries: if the action is invariant under a continuous transformation, there is a corresponding conserved quantity.
• 12:40:00 The connection between symmetries and conservation laws is not just a mathematical curiosity; it explains why the laws of physics look the way they do. If you have a symmetry, you necessarily have a conservation law. Noether’s insight unified a lot of disparate conservation laws under a single framework.

15:29:00 Noether’s First Theorem

Speaker: Derek

• 15:29:00 Time-translation symmetry leads to energy conservation. If you can translate time by a small amount ε and the action remains unchanged, then the quantity dL/dv · v − L (the energy) is conserved. In the simplest Lagrangian with L = T − V, this reduces to the familiar total energy E = 1/2 m v^2 + V.
• 15:59:00 Thus, energy conservation is a direct corollary of time-translation symmetry. Noether’s First Theorem thus explains the origin of all conservation laws associated with continuous symmetries in a static, empty universe.

18:24:00 The Continuity Equation

Speaker: Derek

• 18:24:00 Noether’s Second Theorem concerns local (gauge) symmetries and yields a continuity equation rather than a strict global conservation law. A continuity equation describes the local conservation of a quantity (like energy or charge) by equating the time rate of change of density to the divergence of a flux. In GR, the continuity equation is local: energy is conserved within a small patch of spacetime, but when you try to glue patches together across curved spacetime, the curvature terms modify the equation.
• 18:55:00 The gravitational field itself can exchange energy with matter due to spacetime curvature. As space and time themselves stretch, energy can appear to disappear from one region and “appear” in the gravitational field or in the curvature terms. The continuity equation in GR is thus more complex than the simple global energy conservation in flat spacetime.
• 19:45:00 This is exactly why the total energy of the universe is not strictly conserved in GR once you take the expansion and curvature into account. The Noether continuity equation accounts for the energy that flows into or out of the gravitational field via curvature, which is not captured by a simple energy density term.

23:20:00 Escape from Germany

Speaker: Derek

• 23:20:00 The biography of Emmy Noether: In the 1920s she was a professor at Gottingen, and in 1933 the Nazis banned Jewish people from universities. She was suspended but continued teaching at home and eventually moved to Bryn Mawr College in the United States, where she taught until her death. Einstein called her "the most significant creative mathematical genius thus far produced since the higher education of women began" in an obituary.
• 24:00:00 The story underscores how Noether’s theorems reshaped physics—her work became foundational not only for relativity but for quantum theory and gauge theories that describe fundamental particles and forces. Noether’s two theorems remain central to modern physics and the pursuit of a deeper theory of everything.

24:49:00 The Standard Model - Higgs and Quarks

Speaker: Derek

• 24:49:00 Noether’s insights helped physicists understand why particles have mass and how forces arise from symmetries. In the 1960s–70s, Noether’s theorems contributed to the development of gauge theories, which describe how fundamental forces operate via symmetry principles. The Higgs mechanism and the existence of quarks are tied to these symmetry-based ideas. The video hints that Noether’s work set the stage for the modern Standard Model of particle physics, where symmetries and their breaking give rise to the properties of particles and their interactions.

25:30:00 Closing reflections

Speaker: Derek

• 25:30:00 Noether’s theorems changed the mindset of physics. Physicists began to frame questions in terms of symmetries and conservation laws, applying these ideas across classical and quantum domains. The video closes by highlighting that these theorems are among the most important in the physics of the 20th century and that future explorations will delve deeper into their impact on a theory of everything.

26:30:00 Resources, credits and sponsorships

Speaker: Derek

• The video lists references, biographies, and papers related to Emmy Noether and energy conservation in GR.
• It also acknowledges contributors, sponsors (KiwiCo), and partners involved in animation and research for the video.
• It invites viewers to subscribe and hints at a follow-up video that will cover further implications of Noether’s theorems.

Resources Mentioned

• MathHistory Biography - Emmy Amalie Noether (mathshistory.st-andrews.ac.uk)
• Bartlett, J. G. (2021). The Standard Cosmological Model and CMB Anisotropies. New Astron.Rev. 43 (1999) 83-109
• Kimberling, C. (1982). Emmy Noether, Greatest Woman Mathematician. Mathematics Teacher, March 1982, Volume 84, Number 3, pp. 246–249.
• Blunck, A., et al. (2016). Mathematik und Gender. Verlag Franzbecker
• Evans, R. J. (1976). The Feminist movement in Germany, 1894-1933. Sage Publications
• Without Emmy Noether, there would be a huge gap in mathematics and its understanding (mpg.de)
• Noether, E. (1908). Über die Bildung des Formensystems der ternären biquadratischen Form. Reimer
• Noether, E. (1983). Invariante Variationsprobleme. Gott.Nachr.1918:235-257,1918; Transp.Theory Statist.Phys.1:186-207,1971
• Noether, E. (2018). M. A. Tavel’s English translation of “Invariante Variationsprobleme” (arXiv)
• Byers, N. (1998). E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws. Mod.Phys.Lett. A14 (1999) 99-104
• De Haro, S. (2021). Noether’s Theorems and Energy in General Relativity. Cambridge University Press
• Rowe, D. E. (2019). Emmy Noether on Energy Conservation in General Relativity. arXiv
• Brand, K. (2005). A Note on General Relativity, Energy Conservation, and Noether’s Theorems. Springer
• Additional References: https://ve42.co/NoethersAdRefs

• Directed by Matthew Davies; Written by Matthew Davies, Casper Mebius, Derek Muller and Pablo Rovetto; Edited by Trenton Oliver, Peter Nelson and Nick Lear; Animated by Fabio Albertelli, Emma Wright, Andrew Neet, Alessandro Roussel and Ivy Tello
• Illustrations by Jakub Misiek, Cainejan Esperanza, Tommy Stevens, Maria Gusakovich and Emma Wright; Additional research by Henry van Dyck, Gabe Strong and Darius Garewal
• Music from Epidemic Sound

Visuals to Recreate (described)

• A sequence showing a rock being thrown in space, then slowing due to non-conservative effects (to illustrate energy loss in a non-ideal setting).
• A set of simple triangles demonstrating axes of symmetry (three axes) and rotations (120°, 240°, 360°) with a rotating triangle, highlighting symmetry operations and conservation implications.
• A depiction of a function with a constant added (translation symmetry) and its derivative remaining unchanged, to illustrate the concept of symmetries in functions.
• A spacetime diagram illustrating a local patch of spacetime with a conservation-like equation flowing energy between matter and gravity, to visualize the local continuity equation and curvature contributions.
• A historical timeline showing Noether’s life events (Gottingen, 1933, Bryn Mawr) and Einstein’s esteem for her work.

Notable quotes (verbatim)

• "Imagine you are an astronaut out drifting in deep space when you throw a rock as hard as you can. What's gonna happen to that rock? You would think that it would continue with constant velocity in a straight line. That's just Newton's first law. But what actually happens is it eventually slows down and stops. So why does this happen? Where did all the rock's energy go?" (00:00)
• "Fortunately, he knew just the person for the job. His new assistant, Emmy Noether." (00:02:35)
• "Symmetry is one of those ideas that's easy to recognize but harder to describe." (00:03:40)
• "Time translation symmetry is equivalent to saying that energy is conserved." (18:24)
• "Noether’s two theorems are probably the most important theorems for physics of the 20th century." (01:??)

Visuals (summary of visuals)

• Diagrams illustrating tensors, vectors, and coordinate transformations to show how objects transform under a change of coordinates.
• Animations of spacetime curvature and how energy is exchanged between matter and gravity via curvature terms in GR.
• Historical portraits and archival-style overlays to contextualize Noether and Einstein.

Notes on structure

• The video builds from a basic introduction to symmetry to Noether’s theorems, then to how these ideas apply (or fail to apply) in General Relativity, and finally to Noether’s biography and broader impact on physics.

Additional context

• The discussion connects classical symmetries (space, time, rotation) to conservation laws, then extends to local gauge symmetries and their consequences (continuity equations) in curved spacetime. The core tension is between global conservation laws and the local, curvature-driven dynamics of GR.

Summary of the arc

• Symmetry is the real engine behind conservation laws. Noether showed that continuous symmetries generate conservation laws. Time-translation symmetry gives energy conservation in a static, empty universe; in a dynamic, curved spacetime, local continuity equations replace simple global energy conservation, and the energy attributed to gravity is not a tensor under general coordinate transformations. Noether’s work reframes how physicists think about conservation laws, symmetries, and the foundations of gravity.

Final note

• The video closes by highlighting Noether’s profound influence on modern physics, including gauge theories, the Standard Model, and our understanding of mass and forces, while promising a follow-up that will delve deeper into these ideas.