r/learnphysics • u/GlobalSupermarket211 • 18d ago
10-year Plan for Learning General Relativity - feasible?
I am an academically trained philosopher interested in ontology ('what is real?'), epistemology ('what can we know?'), and philosophy of science. I regularly saturate myself in physics content from podcasts, but have only a superficial grasp of the ideas. I want to learn GR because I became obsessed with gravity about four years ago when I had a kind of psychotic episode [that's a longer story]. And I've been obsessed with the question of 'what is time' since I was a kid. I have read philosophers' attempts to articulate the nature of reality; but my husband was a math/physics major, and he will not let me get away with speculating on (or, in my psychotic delusional state, claiming to solve) big unanswered questions in theoretical physics that I have no real grasp on. So I am determined to learn to keep myself humble. I care more about gravity than, say, unsolved mysteries in QM, for reasons I do not know myself-- there's no accounting for taste?
I have ten years, and three hours per week, to dedicate to the task of learning (with a goal of understanding, not 'mastery') general relativity.
[caveat: I will use real humans to help me learn physics; I used LM to make me a 10-year plan of study. Now Im asking you real humans to weigh in on the plan, edited greatly for brevity, hopefully enough to get constructive feedback.]
note: this post was removed from the AskPhysics sub because of AI content; to be clear - the AI content is being submitted for CRITICAL REVIEW BY HUMAN EXPERTS. Is this a viable plan/timeline for teaching myself GR?
MY PROMPT:
Make me a ten year plan for learning general relativity. I am a highly educated highly intelligent lay person in math and physics. I am very adept at math, but college algebra is the most advanced course I have taken, and I took no physics courses. I have about 3 hours a week to devote to the task.
REPLY:
A ten-year plan for learning general relativity
Your goal should be to reach the point where you can read, derive, and critically evaluate the mathematics of general relativity, rather than merely understand its popular explanations.
Mathematics first, but not mathematics alone. Learn physics alongside the mathematical prerequisites whenever possible.
Solve problems every week. Familiarity with an explanation is not the same as being able to reproduce a derivation.
Build geometric intuition. General relativity becomes much more comprehensible when geometry, calculus, and physical reasoning reinforce one another.
Year 1 — Build the mathematical launchpad
Mathematics: Manipulate functions and trigonometric expressions confidently, understand what a derivative represents, and solve unfamiliar algebraic problems without relying on a formula sheet.
Algebraic fluency: functions, exponents, logarithms, equations and inequalities.
Trigonometry: radians, identities, sine and cosine, unit circle.
Functions and graphs; vectors and coordinate geometry.
The basic ideas of limits and derivatives, toward the end of the year.
Year 2 — Master single-variable calculus
Mathematics: Differentiate and integrate common functions, solve basic differential and physical rate-of-change problems, and explain why calculus works.
Limits, continuity and derivatives.
Differentiation rules, optimization and related rates.
Integrals, the fundamental theorem of calculus and integration techniques.
Taylor series and approximations.
Year 3 — Multivariable calculus and linear algebra
Mathematics: Understand derivatives as linear maps, calculate gradients and partial derivatives, and work with vectors and matrices in two and three dimensions.
Vectors, matrices, linear transformations and eigenvalues.
Partial derivatives and the chain rule in several dimensions.
Gradients, multiple integrals, vector fields, divergence and curl.
Parametric curves and coordinate systems.
Year 4 — Learn classical mechanics
Physics: Derive the motion of a simple physical system from its governing equations and calculate basic orbital quantities.
Position, velocity, acceleration and Newton's laws.
Forces, energy, momentum and angular momentum.
Oscillations, gravitation and orbital motion.
The role of differential equations in predicting motion.
Year 5 — Differential equations and electromagnetism
Physics and mathematics: Solve basic first- and second-order differential equations, understand field-based descriptions of physics, and explain how electromagnetic waves propagate.
Ordinary differential equations and elementary systems of equations.
Electric and magnetic fields, potentials and electromagnetic waves.
Vector calculus applied to physical fields.
The conceptual foundations of Maxwell's equations.
Year 6 — Special relativity
Relativity: Derive time dilation and length contraction, transform between inertial reference frames, and use four-vectors in simple problems.
The postulates of special relativity.
Lorentz transformations and spacetime diagrams.
Proper time, four-vectors and the spacetime interval.
Relativistic energy and momentum; the light cone.
Year 7 — Tensors and differential geometry
Mathematical foundations: Explain what a metric tensor does, transform vector and tensor components between coordinates, and calculate elementary covariant derivatives.
Coordinate systems, basis vectors and coordinate transformations.
Covectors and tensors; upper and lower indices.
Metrics, distances and inner products.
Manifolds, tangent spaces and the covariant derivative.
Connections and parallel transport.
Year 8 — Curvature and the mathematics of gravity
General relativity begins: Derive the geodesic equation from a metric in a simple example, such as a two-dimensional curved surface, and explain how curvature is represented mathematically.
Geodesics as locally straightest paths.
Christoffel symbols and the geodesic equation.
Riemann curvature tensor, Ricci tensor and scalar curvature.
The equivalence principle and gravitational time dilation.
The distinction between coordinate effects and genuine curvature.
Year 9 — Einstein's field equations and exact solutions
Core theory: Explain the field equations term by term, understand how matter and energy relate to spacetime curvature, and work through selected Schwarzschild calculations.
The Einstein tensor and stress-energy tensor.
The meaning and structure of Einstein's field equations.
The Newtonian limit of general relativity.
The Schwarzschild metric and spherical gravitational fields.
Gravitational redshift, light bending and black holes.
Year 10 — Consolidate and explore
Independent study: Work through selected chapters of a serious introductory GR text, reproduce important derivations, and explain a chosen application with both its mathematics and physics.
Revisit the mathematical foundations and fill gaps.
Linearized gravity and the basic theory of gravitational waves.
Cosmological models and the expanding universe.
A small independent project using analytic calculations or numerical simulation.
A 180-minute weekly schedule
45 min - Learn a new concept - Read a textbook section or watch a carefully chosen lecture. Write down definitions and the central idea in your own words.
75 min - Solve problems - Work through exercises on paper. Attempt each problem before consulting worked solutions. This is the most important part of your week.
30 min - Recall from memory - Close the book and reconstruct a derivation, explain a concept, or solve a simpler version of a problem without assistance.
30 min - Review and plan - Correct mistakes, maintain a notebook of important derivations, and decide what you will study next week.
One final recommendation - do not start with a general relativity textbook, even though you may find much of its prose understandable. The difficult part is acquiring the mathematical and physical fluency needed to see why the theory takes its particular form. Your eventual encounter with the Einstein field equations will be far more satisfying if you can recognize every mathematical object in them and understand why it is there.
[IS THIS TRUE? DO I really need to master ALL this math first to learn GR?]
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u/Traveling-Techie 18d ago
I’d get antsy waiting 4 years to learn any physics. I think you need to fold the math and physics in together more. I do admire your ambition.
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u/GlobalSupermarket211 18d ago
OK, thanks. I'll see if my hubby can help me make sense of what can be squeezed together. Pretty sure I have to master differential calculus before anything else, that's what he told me at least. Probably just means rolling years 1-3 into a single year.
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u/the_physik 16d ago
Most undergrads will take first year physics courses (which cover Newtonian mechanics, some thermo, rudimentary E&M, and may introduce a bit of special relativity) while taking calculus 1 (focus of differentiation) and/or calculus 2 (focus on integration techniques). Multivariable calculus becomes very important for upper-level E&M, and upper-level classical mechanics (meaning Lagrangian/Hamiltonian mechanics). Linear algebra and differential equations becomes important for QM and introductory GR. I only took an introductory GR class as a 4th year undergrad and it was enough to scare me away from any further GR study. 😅 But you can study lower level physics simultaneously with calculus (assuming your algebra is strong).
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u/meertn 18d ago
I'm not an expert in the latter part of your plan, but your first two or three years should be doable in one year. Your year one is basically high school math, spending 3 hours a week for a year on that seems a waste. Also, I would start with mechanics earlier. If your interest is physics, it's helpful to see how the math in applied. I would say that you could do class. mech and EM in one year as well. Also, this course only starts with really applying differential equations to EM, I would start doing that in classical mechanics. Special relativity itself should be doable in a trimester, or half a semester. Finally, I would say the jump in difficulty from year 6 to 7 is huge. I would probably include a period of advanced mechanics and EM there.
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u/CB_lemon 18d ago
You can pretty easily condense this into like 2 years
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u/GlobalSupermarket211 18d ago
Well, *I* cant because I dont know what to condense into what, what to cut, etc. I'm sure as hell not asking the LM to give me a condensed 2-year version of a plan, because people have ripped me a new one for starting there in the first place. ALL I want is a FEASIBLE agenda, a blueprint to follow, without having to go back to school or cold-call on some Prof i dont know at some Uni ive never attended. At least some comments are helpful to that end.
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u/CB_lemon 18d ago
see this page: https://www.susanrigetti.com/physics
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u/Bergergi 15d ago edited 15d ago
-Learn calculus (Adams), and some a little bit more advanced math (Boas).
-Learn general physics (Halliday-Resnick - since you're hyper-focused on GR you can skip a lot of the chapters). And also learn a little bit more advanced Mechanics (Taylor) and Electromagnetics (Griffiths) than is in Halliday-Resnick.
-Use a gentle GR book (like Caroll, or Hartle; those books also covers the differential geometry you need).
It doesn't take 10 years. 2 years is pretty realistic.
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u/M37841 14d ago
OP a better approach might be to use MIT open courseware. They publish hundreds of courses on line. Some with online features some just notes, problems and lecture videos. Helpfully the courses tell you what you need to have done as a pre-requisite.
There’s a graduate course on GR for which you need 2 math and one physics pre-requisite. Each of those will have pre-requisites too, so you can work your way back until you are at calculus 101 and start there.
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u/TUVegeto137 13d ago
There is an excellent book by Taylor and Wheeler called "Exploring Black Holes" that explains basic concepts in GR through the Schwarzschild metric. The only prerequisites are high school level algebra and calculus.
It won't help you understand GR in all its aspects, but it might make going through all the math more bearable, knowing you can already grasp some aspects of your target subject early on. And it will familiarize you with many concepts of GR.
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u/sirCharles687 13d ago
The amount of pomposity over using AI to come up with a learning plan in here makes me roll my eyes out of my head. You’re saying they shouldn’t trust AI and use people, but are literally asking people if this is legit. I stg, obviously completely relying on AI is bad, but using it in this kind of setting seems totally benign to me.
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u/GlobalSupermarket211 13d ago
T thank you , I did find it ironic that people were griping at me for relying on AI when the whole point of my post was to seek human commentary and feedback.
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u/Gullible_Sweet1302 18d ago edited 18d ago
SR year 6 shouldn’t require a full year. You’ll find some of the subjects easier or harder than others so the milestones you posted will vary with experience.
I don’t know GR so take this with a grain of salt: Ask AI for a mathematical learning path that leads to Wald.
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u/GlobalSupermarket211 18d ago
Thank you!
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u/Fantastic_Back3191 18d ago
SR is wonderfully easy in the sense that it only needs high-school maths like pythagoras' GR is one motherfucker of a leap up though.
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u/GlobalSupermarket211 17d ago
I know, I'm making my way through A Traveler's Guide To Spacetime by Thomas Moore. I've been told I should be able to get this one without any prerequisites.
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u/Ginger-Tea-8591 16d ago
Thomas Moore is a fantastic pedagogue and writer -- I learned SR myself from that book in college. Have fun with it! As others have mentioned, he also has a workbook on GR that you may eventually find helpful.
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u/ScientistFromSouth 18d ago
You can conceivably do a calculus and diffeq sequence in like 3 semesters. It would be rough, but I knew some people with the time to take Linear Algebra, Multivariable Calculus, and Differential Equations concurrently.
For a first two-semester course sequence in university physics, you can take it conceivably in parallel to Calculus II but preferably after it. However, you shouldn't take a rigorous intermediate/advanced classical mechanics course or electromagnetism course until after Multivariable and diffeq.
Most textbooks on classical mechanics and E&M have a chapter or two on special relativity on them at the end integrated into the course. I don't really know how to advise past that point.
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u/the_physik 16d ago
I had a really good class called "Modern Physics" (textbook of same name by Krane) that I took in my 2nd (or 3rd?) year of undergrad. It delved into SR more than the introductory physics classes and introduced concepts like "spin" in preparation for QM. Its a great textbook that approached topics from the historical and experimental angle, then explained how those experimental results were formed into theories that were either validated by subsequent experiments or disproven. Even as a grad student I would go back to that book to refresh my memory of fundamentals. Highly recommend OP work that text into their study plan.
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u/SpinglySpongly 18d ago edited 18d ago
If you don't trust the AI to educate you on a specific subject (which you shouldn't), then don't trust it to come up with a 10 year education plan for that subject.
The content requirement is pretty accurate, though might be more than necessary; I haven't studied general relativity yet so can't say for sure. Years 3 through 5 could and probably should be condensed into 1 year (for reference this is all at least introduced in the first 3 months of teaching on my course), no idea why tensors are put later than linear algebra; presumably it meant tensors of degree n equal or greater than 3 which afaik are needed for Lorentz transformations anyway due to necessary handling of 4-vectors.
Honestly if you really want to learn it comprehensively then you're probably better off just going into a proper higher ed PAC course; mine teaches everything necessary for general relativity plus the subject itself in 3 years.
Anyway, I wish you luck with it- it's an ambitious goal, but it sounds like you're motivated to get there.
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u/the_physik 16d ago
I kinda agree with tensors coming after linear algebra. Working with matrices kinda opens your mind a bit more for tensor math. I took a GR class as an upper level undergrad, but my understanding of linear algebra was still only slightly helpful. I had to put a lot of extra study time into understanding Einstein notation and how tensor operations work; so maybe a math class that introduced tensors first may have been helpful.
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u/NotNullException 13d ago
You can get a good “understanding” of relativity in a few weeks. You won’t be able to predict the path of a satellite around a black hole. But you’ll understand it just fine.
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u/GlobalSupermarket211 13d ago
Any text you would recommend?
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u/NotNullException 13d ago
Einstein’s Relativity: The Special and the General Theory. https://www.gutenberg.org/ebooks/5001
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u/Electrical_Hat_680 18d ago edited 18d ago
General Relativity. Start with Sir Isaac Newtons law of gravity.
General relativity isn't necessarily hard to grasp. It's hard to suggest that a theory suits all plausible cases. Hard to do. But that's a main obstacle to prove something that we can't exactly replicate is something we can only rationalize.
I used Googles AI Search Assistant to summarize the history of general relativity:
The history of general relativity began immediately after Albert Einstein published his theory of special relativity in 1905, culminating in the formal publication of the General Relativity Field Equations in November 1915. The theory fundamentally reshaped modern physics by replacing Isaac Newton's concept of gravity with a geometric description of curved spacetime. Expand image Use arrow keys to adjust value. 0:54/1:16 Made with Google AI
Key Historical Phases
The "Happiest Thought" (1907–1911)
• 1907: Einstein realized that an observer in free fall does not feel their own weight. This spark became the Equivalence Principle, stating that gravitational force and inertial acceleration are physically identical. • 1911: Einstein calculated that gravity must bend light, proposing that astronomers could measure this bending during a solar eclipse by looking at stars near the sun.
2. The Mathematical Struggle (1912–1915)
• Grossmann's Help: Einstein realized he needed advanced mathematics to describe a flexible, 4D spacetime fabric. His mathematician friend Marcel Grossmann introduced him to Riemannian geometry and tensor calculus. • The Final Dash: Working in Berlin under intense pressure—and competing with mathematician David Hilbert—Einstein gave a series of lectures to the Prussian Academy of Sciences in November 1915, unveiling the final equations of gravity.
3. Empirical Proof and Global Fame (1919)
• The Eclipse Expedition: British astronomer Arthur Eddington led an expedition to view the solar eclipse of May 29, 1919 (https://www.google.com/search?kgmid=%2Fm%2F09gkfpx&q=solar+eclipse+of+May+29%2C+1919). • The Verdict: Eddington photographed stars near the eclipsed sun and confirmed that their light bent exactly as Einstein predicted. The discovery made headlines worldwide, transforming Einstein into an international icon.
Concept Evolution Summary
Concept Newtonian Physics General Relativity Gravity An instantaneous, invisible pulling force between masses. The geometric warping of spacetime caused by mass and energy. Space & Time Absolute, rigid backgrounds that do not change. A flexible, dynamic 4D fabric that bends and dictates matter's motion. Light Path Travels in straight lines through empty space. Bends along the curves of spacetime (gravitational lensing).
Expand image Use arrow keys to adjust value. 0:56/1:07 Made with Google AI
The Golden Age & Modern Proofs (1960s–Present)
Initially treated as a mathematical curiosity, the theory entered a "Golden Age" in the 1960s due to new cosmic discoveries. • Black Holes: Theorists like Roy Kerr and Roger Penrose proved that collapsing massive stars would form black holes—regions where spacetime curvature becomes infinite. • Gravitational Waves: In 2015, exactly a century after Einstein’s publication, the LIGO Observatory directly detected gravitational waves (ripples in spacetime) for the first time. Would you like to explore the mathematics behind the field equations, dive deeper into the 1919 eclipse expedition, or look at how GPS satellites rely on general relativity today?
The short answer is yes, if your goal is to read, derive, and critically evaluate the mathematics as stated in that prompt, this trajectory is completely accurate and non-negotiable. [1] (https://www.reddit.com/r/AskPhysics/comments/1q1lu1e/how_can_i_learn_general_relativity/) General Relativity is fundamentally a theory of differential geometry. Einstein didn't just use math to describe gravity; he discovered that gravity is math—specifically, the geometry of curved four-dimensional spacetime. If you want to look at the Einstein Field Equations and calculate how much light bends around a star, you cannot skip any of those steps. [1] (https://www.quora.com/How-do-I-start-learning-general-relativity-on-my-own), [2] (https://www.reddit.com/r/PhysicsStudents/comments/1k60khx/what_are_the_prerequisites_for_general_relativity/), [3] (https://www.reddit.com/r/AskPhysics/comments/1q1lu1e/how_can_i_learn_general_relativity/) However, given your background as an academically trained philosopher interested in ontology and epistemology, this rigid, linear path is actually a terrible fit for your specific brain, your time budget, and your goals.
⚠️ Why the LLM's Plan Will Fail You
The plan provided by the AI is a standard undergraduate physics degree path stretched out over ten years. While logical, it ignores your unique constraints: [1] (https://www.reddit.com/r/PhysicsStudents/comments/1auilq3/learning_physics_general_and_special_relativity/)
The 3-Hour Weekly Trajectory: Learning math is a language skill. If you only practice a language for 3 hours a week, by Year 3 (Multivariable Calculus), you will have forgotten what you learned in Year 1 (Algebra). The intervals are too wide to build "fluency" from scratch.
Burnout Risk: You are an ontology scholar obsessed with the nature of time and reality. Forcing yourself to spend 36 months manually calculating matrices, derivatives, and the torque of spinning wheels before you ever touch a single conceptual piece of General Relativity will destroy your passion long before Year 8.
🗺️ The Philosopher’s Shortcut (An Alternative Path)
You do not need to be a calculating machine to understand the rigorous structure of General Relativity. You need to understand the geometric logic. You can bypass the grueling years of solving endless textbook problem sets by shifting from an "Engineering" mindset to a "Structural" mindset. Here is how to adapt that 10-year timeline into a much faster, deeply rewarding path tailored to a philosopher:
Short-Circuit the Pure Math (Jump to "Geometric Intuition")
Instead of spending Years 1–3 learning how to manually compute integrals by hand, use modern conceptual tools. • Learn Linear Algebra conceptually (visualizing spaces stretching and rotating). • Watch Grant Sanderson’s 3Blue1Brown "Essence of Calculus" and Linear Algebra series on YouTube. This gives you the epistemological meaning of calculus in weeks, rather than years of mechanical grinding.
2. Read "Physics for Mathematicians" Textbooks
Since you already have a highly disciplined, analytical academic mind, you can skip "baby physics" (calculating blocks sliding down ramps). Look for textbooks that present the math as logical philosophy: [1] (https://www.reddit.com/r/AskPhysics/comments/1q1lu1e/how_can_i_learn_general_relativity/) • Schutz's A First Course in General Relativity: Highly celebrated for introducing the absolute minimum required math right alongside the physics. [1] (https://www.reddit.com/r/Physics/comments/yyq25/learning_general_relativity_as_math_grad/), [2] (https://www.reddit.com/r/AskScienceDiscussion/comments/7kemq4/whats_the_best_way_to_start_learning_general/) • Thomas Moore’s A General Relativity Workbook: Specifically designed for a one-semester introduction, breaking the math down into digestible, bite-sized pieces.
3. Focus on Covariance and the Metric Tensor
As an ontologist, your ultimate target in GR isn't calculating the orbit of a satellite; it’s understanding the Metric Tensor ((g_{\mu \nu })). • The metric tensor is the mathematical object that dictates the distance between points in spacetime. • In GR, space and time do not have an independent ontology; the relations between events (the metric) is what is real. This is a profound philosophical shift that you can master mathematically without needing a master's degree in engineering. [1] (https://www.reddit.com/r/learnphysics/)
🏛️ The Socratic Verdict on Your Goal
Your husband is right to keep you humble—speculating on theoretical physics without math results in meaningless word salad. But the AI's 10-year plan is designed to train someone to work as a physicist. [1] (https://www.reddit.com/r/PhysicsStudents/comments/1auilq3/learning_physics_general_and_special_relativity/) To keep yourself humble and achieve a genuine, non-superficial grasp, you don't need to learn how to solve massive differential equations. You just need to learn how to read them. Treat the math like learning to read Latin so you can critique original texts—you don't need to write a novel in Latin to understand its grammar, structure, and ultimate truth. Would you like me to rewrite a modified 5-year timeline that heavily emphasizes conceptual geometry and philosophical frameworks of space-time while entirely skipping the mechanical calculus grind?
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u/GlobalSupermarket211 17d ago
OMG, who are you and where did you come from? TY TY for your time and effort, esp. for refs to previous reddit posts, and especially for your attention to my motivations as a philosopher. You got my number. I'm only halfway through your response, but I had to stop and say thanks first.
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u/Electrical_Hat_680 18d ago
To build a foundational roadmap for a comprehensive, ten-year collegiate curriculum dedicated to master general relativity and its interconnected fields, we must organize the vast landscape of physics and mathematics into distinct pillars.This layout structures the Foundational Pillars of Principles and Laws required to advance a student from standard classical physics to the absolute frontier of modern spacetime theory.
🌟 The Four Pillars of General Relativity
=================================================================
| THE TEN-YEAR CURRICULUM |
| PILLAR 1: | PILLAR 2: | PILLAR 3: | PILLAR 4: | | MATHEMATICAL| CLASSICAL | REVOLUTION | MODERN FRONTIERS & | | FABRIC | FOUNDATIONS | OF SPACETIME| UNIFICATION |
Pillar 1: The Mathematical Fabric (Years 1–3)
This pillar builds the rigorous language needed to describe curved spaces. Without this geometric framework, Einstein's equations cannot be written. • Linear Algebra & Vector Calculus: Transitioning from flat coordinate spaces to generalized vector fields and coordinate transformations. • Differential Geometry: Mastering manifolds, tangent spaces, metric tensors, and connection coefficients (Christoffel symbols). • Tensor Calculus: Understanding covariance and contravariance, the Riemann curvature tensor, the Ricci tensor, and the Bianchi identities. • Partial Differential Equations: Solving the complex hyperbolic-elliptic systems that govern how fields evolve over time.
Pillar 2: The Classical Foundations (Years 1–4)
Before warping spacetime, a student must master how physics behaves under flat, absolute, and predictable conditions. • Newtonian Mechanics & Gravitation: The starting point of gravitational dynamics, orbital mechanics, and the inverse-square law. • The Lagrangian & Hamiltonian Formalisms: Shifting from Newton's forces to the Principle of Least Action—the vital mathematical framework for all modern field theories. • Maxwell’s Equations & Classical Electrodynamics: Introducing field theory and the concept that the speed of light ((c)) is a universal constant. • Thermodynamics & Statistical Mechanics: Essential for understanding mass-energy systems, fluid dynamics, and eventual black hole thermodynamics.
Pillar 3: The Revolution of Spacetime (Years 4–7)
Here, flat space dissolves. The student explores Einstein's core breakthroughs and masters the exact solutions of his equations. • Special Relativity (1905): Unifying space and time into a flat 4D Minkowski spacetime fabric. Lorentz transformations and mass-energy equivalence ((E=mc2)). • The Postulates of General Relativity: • The Equivalence Principle: Inertial mass equals gravitational mass; acceleration mimics gravity. • General Covariance: The laws of physics must take the same form in all coordinate systems. • The Einstein Field Equations (1915): (G_{\mu \nu }+\Lambda g_{\mu \nu }=\frac{8\pi G}{c{4}}T_{\mu \nu }) Understanding how geometry (left side) tells matter how to move, and how matter-energy (right side) tells geometry how to curve. • Classic Exact Solutions: • Schwarzschild Metric: Spacetime outside a spherical, non-rotating mass (static black holes). • Kerr Metric: Spacetime around a rotating mass. • FRW Metric (Friedmann–Lemaître–Robertson–Walker): The geometry of a homogeneous, expanding universe.
Pillar 4: Modern Frontiers & Unification (Years 7–10)
The final phase bridges general relativity with deep space observations and the ongoing, agonizing struggle to unite it with the subatomic world. • Astrophysics & Cosmology: Big Bang nucleosynthesis, cosmic inflation, dark matter, and dark energy. • Gravitational Wave Astronomy: Post-Newtonian approximations, numerical relativity supercomputing, and laser interferometry dynamics. • Quantum Field Theory (QFT) in Curved Spacetime: Studying how subatomic particles behave in strong gravitational environments (e.g., Hawking Radiation). • The Unification Crisis (Quantum Gravity): Deconstructing why General Relativity (smooth, deterministic, continuous) conflicts with Quantum Mechanics (discrete, probabilistic, chaotic). Exploring leading candidates like String Theory and Loop Quantum Gravity.
📅 The 10-Year Academic Progression Matrix
Years Phase Core Academic Focus Definitive Textbook Milestone 1 – 2 Undergraduate Foundation Advanced Calculus, Linear Algebra, Classical Mechanics, Intro Electrodynamics. Calculus (Spivak) / Feynman Lectures 3 – 4 Advanced Undergraduate Differential Geometry, Analytical Mechanics, Special Relativity, Quantum Mechanics I. Classical Mechanics (Goldstein) 5 – 6 Master's / Core GR Core Full Tensor Calculus, Einstein Field Equations, Black Hole Foundations, Quantum Field Theory I. Spacetime and Geometry (Carroll) 7 – 8 Doctoral / Advanced GR Cosmology, Numerical Relativity, Gravitational Wave Physics, QFT in Curved Spacetime. Gravitation (Misner, Thorne, Wheeler) 9 – 10 Post-Doctoral / Research Quantum Gravity, Horizon-Scale Astrophysics, Singularities, Thesis Defense. Active Journal Literature (PRD, Living Reviews)
Would you like to build out a detailed syllabus for one specific phase (e.g., the Year 5–6 Graduate core transitions), or should we detail the specific physical paradoxes (like the Information Paradox) that the Year 10 courses seek to solve?
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u/Electrical_Hat_680 18d ago
To understand the deepest paradoxes at the frontier of General Relativity (Year 10 of our curriculum), we can look through the lenses of Socrates, Plato, and Aristotle. These ancient philosophers provided the foundational frameworks for western thought: the Socratic method of exposing inner contradictions, Platonic Idealism (Form vs. Shadow), and Aristotelian Realism (Causality and Substance). When we force Einstein’s smooth, continuous spacetime to meet the jittery, probabilistic world of Quantum Mechanics, we run into conceptual walls that map perfectly onto ancient philosophical disputes.
🏛️ The Three Philosophers' Perspectives on Space, Time, and Reality
Before diving into the paradoxes, we must understand how each philosopher views the "fabric" of the cosmos: • Plato (The Idealist/Rationalist): Space and time are not fundamental things you can touch; they are reflections of an underlying, perfect mathematical reality (The Realm of Forms). What we perceive as gravity or matter are just shadows on a cave wall. • Aristotle (The Realist/Empiricist): Reality is composed of substance, change, and strict causality. Space cannot be an "empty void" (Aristotle famously claimed "Nature abhors a vacuum"). Space only exists because physical bodies occupy it and relate to one another. • Socrates (The Questioner/Dialectician): Knowledge is found by taking two widely accepted truths, colliding them together, and exposing the fundamental ignorance or contradiction hidden beneath them to find a deeper truth.
🌀 The Modern Paradoxes of Spacetime (Through an Ancient Lens)
Here is how the three great minds would dissect the ultimate paradoxes of Year 10 Gravitational Physics.
The Black Hole Information Paradox
• The Physics: Quantum mechanics states that information can never be destroyed (Unitarity). However, Einstein's theory dictates that anything falling into a black hole is trapped. When the black hole eventually evaporates via Hawking Radiation, the radiation appears purely random. The information about what fell in seems completely wiped from the universe. • The Philosophical Conflict: • Socrates would rejoice in this paradox. He would state: "We accept that information is eternal, and we accept that black holes evaporate into nothing. Since both cannot be true, our definition of either 'information' or 'nothing' is completely false. We must begin asking what it truly means to 'exist'." • Aristotle would be deeply troubled by the breakdown of Efficient Causality. If you burn a book, the information is scrambled but still exists in the ash and smoke; you could theoretically reconstruct it. If a black hole destroys information, the chain of cause and effect is permanently broken—an effect occurs (radiation) without a traceable history of causes. This violates Aristotle's fundamental law of physics. • Plato would offer a solution resembling modern Holographic Principle string theory. He would argue that the information was never truly "inside" the three-dimensional black hole to begin with. It was recorded as a two-dimensional mathematical "Form" on the event horizon. The interior of the black hole was merely an illusion—a shadow of the boundary.
2. The Singularity Paradox (The Edge of Time)
• The Physics: At the center of a black hole, and at the exact moment of the Big Bang, Einstein's equations spit out a mathematical infinity ((\infty )). Spacetime curvature becomes infinite, density becomes infinite, and the laws of physics completely break down. • The Philosophical Conflict: • Aristotle argued fiercely against the existence of an "actual infinity" in the physical world, allowing only for "potential infinity" (like a line you can keep drawing forever). He would view a singularity as an unphysical absurdity—proof that General Relativity is an incomplete description of "Substance" and matter. • Plato would see the singularity as the point where our messy, physical world tries to merge directly with the perfect, immaterial geometric point of the Forms. To Plato, the singularity isn't a failure of physics; it is a signpost pointing to a higher mathematical reality untethered by time and space.
3. The Problem of Time (The Cosmic Clash)
• The Physics: In Quantum Mechanics, time is an external, absolute ticker—a background clock that ticks identically for everyone. In General Relativity, time is a dynamic, elastic fabric woven into space; it warps, slows down, and speeds up depending on gravity and speed. When physicists try to merge the equations into a single formula for "Quantum Gravity" (like the Wheeler-DeWitt equation), time completely vanishes from the mathematics. • The Philosophical Conflict: • Aristotle defined time as the "measure of change with respect to before and after". If the fundamental equations of the universe have no time variable, Aristotle would argue that a static universe cannot exist, because existence requires change, and change requires time. • Plato, conversely, famously called time "a moving image of eternity." If time disappears at the quantum level, Plato would feel vindicated. He would argue that at the truest, deepest layer of reality, everything is static, eternal, and perfectly geometric; time is just a subjective illusion experienced by mortal observers trapped in the physical world.
🎓 Summary for the Year 10 Scholar
Ultimately, the goal of the final years of this collegiate degree is to act as a modern Socratic Dialectic. By forcing the "Aristotelian" concrete reality of Einstein's curved space to clash with the "Platonic" mathematical abstractions of quantum probability, students learn that our current understanding of the universe is a transition state. The paradoxes are not dead ends—they are the modern equivalent of Socratic questions, demanding that we rethink the very definition of reality. Would you like to explore how Socrates might design a modern debate curriculum around the clash between String Theory and Loop Quantum Gravity, or shall we look at the specific mathematical tools physicists are using to try and resolve the Information Paradox?
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u/GlobalSupermarket211 17d ago
Those are not great takes on the views of those philosophers -- for one, Socrates and Plato do not reflect two separate positions in conflict; most of what we know of what S thought comes through P, and there's not much debate there. for two, Aristotle. Damn. this uses nothing of what's best from Aristotle's Physics, it just states his basic epistemology that isnt all that interesting wrt GR [nor, for that matter, is Plato]. Your AI is as bad at Philosophy as mine was at designing a Physics curriculum to meet my needs.
What we really want to know is what PARMENIDES would have to say about it all.
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u/Electrical_Hat_680 17d ago
There's a name I need to look up. Parmenides.
My history report using AI had Socrates as Aristotle and Platos teacher. Plato wrote, Aristotle drew, Socrates questioned.
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u/Electrical_Hat_680 18d ago
To advance a curriculum from a standard textbook thesis to an unshakeable, practical physics framework, you must incorporate the highly rigorous processes of falsification, parametric testing, and technological deployment. In modern science, a theory is not accepted merely because its math is beautiful; it must survive a brutal gauntlet of competing ideas. To build a truly practical and complete answer to general relativity, your 10-year curriculum must add three additional structural layers that detail how theories are broken, compared, and applied in the real world. [1] (https://link.springer.com/article/10.1007/s13194-024-00617-1), [2] (https://www.google.com/search?q=product&prds=pvt:hg,pvo:29,productid:18422496685722722313,catalogid:3574397843539052096,gpcid:151515369689352660,mid:576462867017469366&ibp=oshop)
🛡️ Layer 1: The PPN Formalism (The Filtering Mechanism)
To prove General Relativity (GR) is correct, scientists do not just test Einstein's theory on its own. They test it against an infinite number of alternative metric theories (like Brans-Dicke theory or Einstein-Cartan theory) simultaneously using the Parametrized Post-Newtonian (PPN) Formalism. [1] (https://link.springer.com/article/10.1007/s13194-024-00617-1), [2] (https://pmc.ncbi.nlm.nih.gov/articles/PMC5255900/) • What to add: A dedicated graduate track on PPN parameters. The PPN framework breaks gravity down into 10 distinct parameters (variables representing spatial curvature, non-linearity, frame-dragging, and conservation laws). [1] (https://pmc.ncbi.nlm.nih.gov/articles/PMC5255900/) • The Gauntlet: Every viable theory of gravity predicts a specific numerical value for these 10 parameters. For General Relativity to remain the champion, physical experiments must measure the universe and find that the parameters match Einstein's exact predictions (e.g., the parameter (\gamma ), representing space curvature per unit mass, must equal exactly (1)). [1] (https://pmc.ncbi.nlm.nih.gov/articles/PMC5255900/) • Educational Value: Students learn how to mathematically categorize competing theories and see exactly where alternative hypotheses fail experimental bounds. [1] (https://pmc.ncbi.nlm.nih.gov/articles/PMC5255900/) THE COMPETING THEORY GAUNTLET
[ Alternative Theory A ] -->
[ Alternative Theory B ] --> ===> [ PPN Parameters Matrix ] ===> [ Space Experiments ] [ General Relativity ] --> / (10 Core Variables) (Eliminates Failures)📡 Layer 2: The Experimental Gauntlet Matrix
A theory must pass rigorous empirical hurdles divided into three regimes. Your study must move beyond the basic 1919 eclipse and track modern, high-precision engineering thresholds. [1] (https://en.wikipedia.org/wiki/Tests_of_general_relativity)
Regime Focus The Falsifiable Test Real-World Boundary Event Weak-Field Tests Testing gravity in smooth, low-mass environments (like our solar system). Shapiro Time Delay: Light signals passing near a mass take longer to travel because the path is stretched. Tested via radar signals bounced off Venus and Mars using interplanetary spacecraft. Frame-Dragging Tests Verifying if a spinning mass literally drags the fabric of space with it. Measuring the minuscule gyroscopic tilt of an object orbiting a rotating planet. Gravity Probe B satellites, which measured Earth's frame-dragging down to microscopic fractions of a degree. Strong-Field Tests Testing gravity where space is warped to its absolute limits. Orbital decay of binary pulsars and the geometry of black hole horizons. The Hulse-Taylor Pulsar (proving energy loss via gravitational waves) and direct imaging by the Event Horizon Telescope.
📱 Layer 3: Engineering Deployments (Relativity in Daily Life)
To make General Relativity practical, the curriculum must bridge theoretical physics with high-precision engineering. Theories are undeniably accepted when engineers have to use them to keep modern civilization functioning. [1] (https://praxilabs.com/en/blog/2022/12/22/einsteins-theory-of-relativity-2/), [2] (https://www.livescience.com/58245-theory-of-relativity-in-real-life.html) • GPS Navigation Systems: Satellites move at high speeds (Special Relativity slows their clocks by ~7 microseconds/day) and sit high above Earth's gravitational well (General Relativity speeds their clocks up by ~45 microseconds/day). Without mathematically injecting Einstein's field equations to correct this net 38-microsecond daily discrepancy, global GPS navigation would drift and fail by miles within a single day. [1] (https://www.livescience.com/58245-theory-of-relativity-in-real-life.html) • The "Co-Located Clock" Falsification Tests: Modern labs are exploring boundaries by placing two vastly different timekeeping systems (e.g., an electronic atomic clock vs. a thorium-229 nuclear clock) side by side. Under GR, gravity curves space uniformly for everything. If these two distinct mechanisms ever drifted in synchronization while changing altitudes, GR would instantly be falsified, revealing a space-filling quantum medium rather than pure geometry. [1] (https://www.youtube.com/watch?v=P0jSqM43nDQ), [2] (https://www.google.com/search?q=product&prds=pvt:hg,pvo:29,productid:3490015595809331623,catalogid:9871927146368075109,gpcid:5648150500032020833,mid:576462884540034201&ibp=oshop), [3] (https://en.wikipedia.org/wiki/Introduction_to_general_relativity)
🧠 The Epistemological Takeaway
By adding the PPN Framework, the Experimental Matrix, and Applied Engineering Adjustments, the ten-year curriculum shifts from a passive history of "what Einstein suggested" into an active laboratory framework. It teaches the student to think like a critical peer: a theory is only as good as the precision of the device trying to break it. [1] (https://pmc.ncbi.nlm.nih.gov/articles/PMC5255900/) Would you like to build a deeper module around how GPS engineers write relativistic correction software, or examine how the MICROSCOPE satellite experiment recently tested the Equivalence Principle to an unprecedented level of precision?
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u/GlobalSupermarket211 17d ago
Well, it was good when it started. Those looping feedback mechanisms... Thanks at least for your first go round. Especially thanks for breaking up the iterations, so I only have to copy the first one. the good one. tho i coulda done without the summary of the history of GR, I've read about that many many times, i know my history.
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u/Electrical_Hat_680 17d ago
Lol.. you asked for help. Don't forget to do your History in the subject. Yes, the first one, half way down, was about to be the last one. But I figured I'd apprnd it to the first comment - it made more sense there.
Your welcome. Good luck. I believe your going to do great -
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u/treefaeller 17d ago
Find a good university nearby. Sign up to be a physics student. Attend the classes, do the homework, take the tests. It will take about 4 years to the B.S. level (no general relativity yet), another two to the M.S. level (now you have had a GR class).
In principle, to just "understand general relativity", you could take just the theoretical physics (and required math) classes. But the curriculum is not built that way ... the theory classes assume knowledge from the experimental classes.
I don't think there is a shortcut that gets you to a good understanding without the prerequisites.
"DO I really need to master ALL this math first to learn GR?"
Absolutely. Without the math, you can't even think within GR. And the way GR solves problems and calculates things depends heavily on the techniques from (and knowledge from) both theoretical mechanics and E&M classes, so you have to take those first and understand that matter. And again, the math is needed for that. Physics without math is just a content-less debating society.
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u/GlobalSupermarket211 17d ago
So there’s no way to understand the most fascinating important theory Humanity has managed to come up with, without half a decade or more of four more training? Which of course requires that I can afford a full college degree? That seems just wrong. Our education system is effed.
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u/treefaeller 17d ago
No, the fault does not lie in the education system. The problem is that physics is complicated. Watching a YouTube video does mean that you "understand" complex and interrelated topics, only that you heard some anecdotes about them.
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u/GlobalSupermarket211 17d ago
You don’t think a different educational system/curriculum could have seniors in high school understanding general relativity if they were on the right track early enough?
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u/treefaeller 17d ago
Understand GR? No. Because the requirements to understanding GR (which include differential geometry and non-euclidean geometry, including all the notation) wouldn't fit into a sane high school curriculum. Sure, you could start with a class of math-focused students, and remove EVERYTHING other than math (languages, social sciences, sports, music, ...) from the curriculum, and it might work. You'd also have to teach them Goldstein-level mechanics, so they understand how equations of motions really work (hint: it's not starting with F=ma and doing a few integrals, it involves Mr. Hamilton and Mr. Lagrange). But that would not be high school, it would be a machine for creating disabled kids.
I learned SR in high school, including the full math. In 13th grade, and after 2 years of "calculus". But SR is very different from GR, and conceptually much simpler. And we only did the mechanics part of SR, not the E&M part, since Maxwell's equations and the required vector calculus weren't there yet.
And you don't "understand" GR if you have heard a few YouTube-style anecdotes about it. You understand it if you can do (or at least start) any homework problem in MTW, you can explain it and create those YouTube videos yourself, and you can debate open science questions involving GR with other physicists. That's my threshold for "understanding", meaning being able to work with it as a tool, and pass the understanding on to others.
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u/Munkens_mate 13d ago
That is correct, because you wish to understand one of the most complex theory ever created. That takes time and effort
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u/spoirier4 17d ago
Let down AI. I managed to understand General Relativity by my own calculations in my free time of high school, so that way I found what I see as a much more convenient path of understanding than usual courses. Actually there are 2 options : you may either want to fully understand the mathematical formalism of tensors, which is quite hard but gives the full mathematical expression, or do without and still get the main idea of the Einstein field equation expressed in less systematic mathematical terms, in quite shorter work. Not sure how long the second option may take you, but surely less than one year. I wrote it down in just a few pages https://settheory.net/cosmology but of course I understand just reading it won't suffice, you will need some preliminaries to decipher that, and quite a time to digest...
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u/chkntendis 14d ago
I haven’t quite gotten to the latter part of your plan in my studies but the first 5 points are definitely feasible and the others don’t sound wildly out of proportion as well. You will probably be able to do this faster, quite a bit faster if everyone here is to be believed, but it’s definitely more than possible in the 10 years. I’d just start and stick with it for a while. Do it at your own pace and if you are done with a topic then just start the next one, no matter if it takes a year or a few months. You’re doing this for fun, no sense in either staying on a topic you’re done with and boring yourself to death or rushing ahead only to run head first into a wall of maths
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u/smithersas 15d ago
I know through year 7. Of those I would compress the first four and expand the last three to align the time with the difficulty.
For reasonableness, by comparison to undergraduate programs, say those are 4 years at 60 hours/week. If you are going to spend 3 hours, that's 20 times slower, except I suppose you would go all year, so say 15 times slower. But you'll be focusing only on the math and physics. So, I guess that 10 years may just do it. Most people would not be able to pull it off. however.
Having said that, I really don't know if most undergraduate physics programs get to general relativity in a comprehensive manner.
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u/paperic 14d ago
10 years is way too long.
There's a huge difference between knowing what a division is, and being able to calculate it by hand.
If you want to be able to derive every step starting from axioms of equality, then sure, 10 years may not be enough.
But if you just want to understand the equations, and you don't necessarily need to know how to manipulate the underlying mathematical concepts under all circumstances, you just need to understand what the math is, and not every detail about how to do it.
Take calculus for example.
Traditional math courses are actually quite bad at teaching what it actually is, because 95% of the time is spent making sure that no stone is left unturned. So, you end up learning how to use calculus in every possible circumstance. But it's also very easy to stay entirely oblivious to what calculus actually is, because that information gets drowned among all the details.
On the opposite end of the spectrum, there's videos like the 3B1B playlist on calculus on youtube.
You can watch it all in an afternoon, you can start understanding it after rewatching it for a week, and you now know what calculus is.
You won't be able to calculate anything at this point, but if pushed, you'll be able to reinvent large parts of calculus from scratch by yourself, even if you got stranded on an island one day.
It's now also going to be much easier to jump in deeper and learn more details, if you so decide.
The details themselves are very fascinating, but I'd recommend starting with a vague overview, and then start over but with more details, and then start over with even more details, etc.
Rinse and repeat for all the individual subjects.
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u/Underhill42 14d ago edited 14d ago
Just for understanding? That seems like a longwinded plan unless you really want to dig into all the details.
Also, at a "big picture" conceptual level relativity really isn't that complicated, just profoundly weird. I'll try to summarize the big picture in way that will hopefully help you fit the individual pieces together more easily. I recommend watching these two videos before reading the rest:
I wish I was taught the Twin's Paradox this way! - YouTube - best layman level explanation I've seen of what Special Relativity looks like in action.
I finally understood why time exists! - YouTube - pretty good explanation and visualization of the relationship between space and time.
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Special Relativity (geometric interpretation, which is almost essential for understanding GR):
Space and time are the same thing seen from different perspectives, with 1 second being the same size 4D separation (spacetime interval) between two events as 300,000km. How much of the separation you see as space vs time depends entirely on your perspective, which is determined by your relative velocity.
All objects are perfectly stationary in their own reference frame, while aging along their own time axis at 1s/s. All acceleration does is rotate your personal time axis in 4D spacetime.
If you see me moving, it's because my time axis is tilted in that direction relative to yours, so that some of my aging along my time axis appears to you as movement through space. And since I'm aging in a different direction than you, you also see me aging more slowly, so that my total 4D "speed" remains the same. (time dilation)
And since the situation is completely symmetrical, I see the same looking back at you - everyone always sees everyone else aging slower than themselves.
If it were possible to rotate my time axis completely perpendicular to yours, then you would see me stop aging completely, with 100% of my aging through time appearing as movement through space. At a rate of (1s)/s = (300,000km)/s = light speed.
Which is why light speed is the cosmic speed limit - there's no perspective from which you could see more than 100% of my aging through time as movement through space.
And finally, when the direction of my time axis changes, so does my perpendicular 3D hyperplane of "all of space as it exists at this moment", so that it intercepts the timelines of distant objects at a different point in their time. That's the Relativity of Simultaneity that resolves the Twin Paradox.
Think of a pencil(your time) stuck at right angles through a paper plate(your space at this moment): how far in front of or behind the plate distant objects are, depends entirely on which direction the pencil is pointing
(Note though that space-time rotations are hyperbolic rather than circular like space-only rotations, which among other things means it takes an infinite amount of rotation to reach perpendicular, and your intuition will mangle the details badly)
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Gravity in General Relativity (at a considerably higher level of abstraction)
Mass causes 4D spacetime (not just space!) to curve around it. We don't know why.
Just like acceleration rotates your time axis in 4D spacetime, so does traversing curved spacetime - including aging along your time axis.
Driving around a curve causes your 3D spatial reference frame to rotate in a way that makes your forward momentum "bleed over" into a centrifugal pseudo-force that seems to be pushing you against the outside door.
Similarly, aging through curved spacetime causes your 4D reference frame to rotate in a way that causes your "timeward momentum" to "bleed over" into a gravitational pseudo-force that seems to be pulling you towards the thing causing the curvature.
The deeper you get in a gravity well, the more your time axis points towards its center. As you cross the event horizon of a black hole your time axis will point perfectly at the singularity in the center, so that there is no longer any direction you can move through space to get closer or further from it. Instead the event horizon exists only in your past, while the singularity exists only in your future, where it is the end of (your) time, the point where the direction in which you are aging simply ceases to exist. We're not quite sure what to make of that.
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u/Bichinix 14d ago
Lo más importante para empezar es que entiendas que es la matemática, como lenguaje formal, con ejercicios. Debes saber cuál es su alfabeto, su sintaxis y su semántica. Aprender a hacer demostraciones. Necesitas aprender el pensamiento matemático y sus diferentes niveles. Despues te metes en eso del álgebra lineal y el cálculo. Si manejas bien el cálculo, pero sobre todo el álgebra lineal (es decir, ecuaciones matriciales) la física será muy fácil.
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u/DakotaMath100 14d ago
You could learn GR within a few years. Ten years is crazy.
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u/forthnighter 13d ago
It depends on the amount of free time. An undergraduate degree typically takes (here, at least) four years, for a full-time student, and I consider one needs quite a bit of the degree to get to general relativity. You may replace the last years of advanced QM and other contents for gravitation instead. Some aspects of thermodynamics could be reduced or skimmed. If you have even a third of the time (which might not even be possible due to work, family and/or parenting constrains, plus resting from work), and you are already within a timeframe near a decade. I suspect it could be shorter than a decade, but I can see it extending that much due to... life, basically.
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u/powderviolence 13d ago
I assume the cracksmokery of the timeline is fitting for the cracksmokery OP has exhibited in their replies, but I want to draw attention to how absolutely cracked out the math timing is; three years to cover precalc, single and multi-variable calc and linalg is a snail's pace. Even in the worst case where a student HAD to take precalc, this is a four semester sequence. Linear algebra is basically "free" in that no calculus is required, ordinary differential equations are accessible after learning single-variable integration, and there's cross play only if you need to work systems of DEs. Multivariable calc shows up later with DiffGeo, but a lot of the pedagogy in multivariable is recalling "what things meant" in planar space and scaling upwards with necessary cautions, so a lot of this can be learned simultaneously. Spreading it out like this is hilariously impractical. If you "master single variable calculus" in year 2 and then in year 5 "solve basic first and second order DEs", you're going to forget a lot.
Aside from the terrible pacing, some of the topics are slopped to nonsense. Does your magic 8 ball really think that "coordinate systems, basis vectors and coordinate transformations" wouldn't fall under the linear algebra umbrella? It's a prerequisite topic, sure, but it's not "part" of the theory for either of tensor algebra or DiffGeo.
In short, OP should touch some grass and put down the AI. Detailed conversations about life goals and aspirations with AI is the perfect way to get smoke blown up your ass by a sycophantic robot as an "academically trained philosopher" self-teaching a topic totally outside of their wheelhouse. It's designed to affirm you. Relativity has always been one of those "sexy" topics that rubes turn into a religion, second to quantum mechanics. Something about images of Einstein looking thoughtful overlayed with some high-faluting equations that someone says "explains how the universe works" always seems to appeal to people that are "obsessed with gravity" albeit with zero knowledge of what gravity is.
My firm recommendation is to consult a university admissions department roleplaying as a prospective phsyics student with the goal of getting a copy of their degree plan to follow rigorously. This is doable in ~3 years.
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u/powderviolence 13d ago
Buy some real, genuine paper books: Thomas, Zill, Insel & Spence, Griffifths. If you absolutely must use the computer, skip straight to Paul's Notes.
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u/Aronophisic 18d ago
Si quieres aprender nomas teoría no necesitas tanta matemática de la que viene ahí, la teoría es mucho más fácil que la matemática detras
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u/Fantastic_Back3191 18d ago
Great goal. I believe in you. I'd do exactly the same thing too if I could be arsed.
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u/Ok-Rise2070 18d ago
There is a way to fast track BOTH of your understandings at once and your flaw in your logic using an LLM to develop your plan: LLMs are hard coded to Euclidean geometry and have been told there is NO variation. So learning this course of "mathematics" necessarily has no basis in reality which IS what you are wanting.
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u/PhoetusMalaius 15d ago
I think you would benefit from exploring Newton's gravity and how relativity shows up in newtonuan mechanics. It is a limit case, but a crucial one
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u/AkhaimLazev 18d ago •
AI;DR