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