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Galactic Rotation Curves etc.

The problem of galaxy rotation curves. The rotational speed of stars does not decrease along the radius, meaning that even at large radii, stars still have high rotational speeds. However, the solar system conforms to Newton's laws

Physical laws should maintain the same form in any coordinate system, meaning they should have general covariance. The Poincaré group only applies to gravity-free flat spacetime (i.e., Minkowski spacetime). In the presence of gravity, more general manifolds and metric tensors must be introduced to describe the curvature of spacetime

General relativity: A simplest metric theory
Any physical theory written in a particular coordinate system can be re-expressed in coordinate-independent geometric language
Mach's principle: All motion is relative. The inertial force experienced by an object is the combined effect exerted by the acceleration of other matter in the universe relative to it
Local Lorentz invariance: The results of any local non-gravitational experiment are independent of the velocity of the (freely falling) apparatus

Translation transformations are affine transformations (Affine Transformation)
In a sufficiently small freely falling reference frame, physical laws are consistent with those in gravity-free Minkowski space
However, on a larger scale, the influence of gravity causes relative transformations between local inertial frames to no longer be Poincaré transformations but more general nonlinear coordinate transformations
The Poincaré group is a finite-dimensional Lie group, while the diffeomorphism group is an infinite-dimensional Lie group
Their relationship involves the generalization from rigid symmetries to differentiable transformations on general manifolds

Manifold: Locally looks like \(\mathbb{R}^n\)
The differential structure of a manifold ensures invariance under coordinate transformations (smooth transitions between different observers)
Although a manifold can be locally regarded as Euclidean space, it generally lacks a global linear structure
The value of a tensor field at a point is called a tensor
Ordinary matrices can be considered as (1,1) type tensors. The contraction of tensors corresponds to taking the trace of a matrix
The direct product of matrices corresponds to the direct product of tensors: the product of matrices corresponds to the direct product of tensors followed by contraction

Line element: On a differentiable manifold M, take a non-degenerate, symmetric, second-order covariant tensor field to construct a quadratic form
If \(\det(g_{μν})=0\), then the metric is degenerate, indicating that measurement along certain directions disappears (e.g., the existence of nonzero vectors with zero length)
Non-degeneracy means no loss of information and prevents different entities from becoming indistinguishable

A photon experiences zero time in its own reference frame
The Schrödinger equation transforms into the diffusion equation of classical statistical physics after Wick rotation
Although the imaginary-time method efficiently solves the ground state, recovering real-time evolution from it remains difficult

Anti-de Sitter space can be viewed as embedded in a higher-dimensional Minkowski space
Extra dimensions must be extremely small
Berry curvature is like the gravitational field of Hilbert space

The effective charge \(e(E)\) in QED increases.
At an extremely high energy scale \(E_{Landau}\), the charge tends to infinity, leading to the Landau pole

The rotation of spacetime itself, i.e., frame dragging (Frame Dragging)
Ergosphere (Ergosphere)
The wave version of the Penrose process extracts additional energy from the rotation of a black hole and reflects it outward

Massive black holes (such as the black hole at the center of the Milky Way) have evaporation times far exceeding the age of the universe (more than \(10^{60}\) years), meaning they will not disappear in the observable future
Small-mass black holes (such as microscopic primordial black holes) may have evaporated entirely in the early universe
When a black hole's mass becomes extremely small (approaching the Planck scale), its temperature rises sharply and may end in a final explosion, emitting high-energy gamma rays. However, this remains an open question. Hawking radiation has not been directly observed because, for astrophysical-scale black holes, their temperature is lower than the CMB, making detection nearly impossible