# 查看完整版本 : 張量分析,狹義相對論

Goodman1945 2023-4-1 22:01

topochu 2023-4-5 13:24

lightspeed2020 2023-4-13 16:55

[quote]原帖由 [i]Goodman1945[/i] 於 2023-4-1 22:01 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=557220548&ptid=31061597][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

lightspeed2020 2023-4-17 01:10

1. {x^μ} = {t, x, y, z} = {x^0, x^1, x^2, x^3} = {x^0, x^i}
2. {e_μ} = {e_t, e_x ,e_y ,e_z } = {e_0,e_1,e_2,e_3} ={e_0,e_i}
3. e_µ · e_ν =η_µν
4. η_µν = diag (-1,1,1,1)
5. η^µνη_νρ = δ(^µ)(_ρ)

Lotus6268 2023-4-17 18:27

[quote]原帖由 [i]Goodman1945[/i] 於 2023-4-1 22:01 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=557220548&ptid=31061597][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

topochu 2023-4-18 18:22

[[i] 本帖最後由 topochu 於 2023-4-20 22:38 編輯 [/i]]

topochu 2023-4-18 18:24

[quote]原帖由 [i]Lotus6268[/i] 於 2023-4-17 18:27 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=557625960&ptid=31061597][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

lightspeed2020 2023-4-20 16:57

[quote]原帖由 [i]Goodman1945[/i] 於 2023-4-1 22:01 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=557220548&ptid=31061597][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

What are the challenges in solving the highly coupled and nonlinear Einstein field equations in general relativity, and what are the early successes in solving these equations for cases with high symmetry, such as the Schwarzschild solution for a vacuum spherically symmetric black hole and the Friedmann-Lemaître-Robertson-Walker metric for describing the evolution of a homogeneous and isotropic universe?
The Einstein field equations in general relativity are a set of ten highly coupled and nonlinear partial differential equations that relate the curvature of spacetime to the distribution of matter and energy. The complexity of these equations makes their solution difficult, and until recently, only a few solutions were known. In particular, early successes in solving these equations were for cases with high symmetry, where the equations simplify due to the presence of symmetry.
The Schwarzschild solution is an exact solution to the Einstein field equations for a spherically symmetric, non-rotating black hole in a vacuum. The solution is given by the metric tensor:

[attach]14204741[/attach]

where G is the gravitational constant, M is the mass of the black hole, c is the speed of light, r is the radial distance from the center of the black hole, and t, \theta, and \phi are the usual spherical coordinates.
The Friedmann-Lemaître-Robertson-Walker metric is an exact solution to the Einstein field equations that describes the evolution of a homogeneous and isotropic universe. The metric is given by:
[attach]14204742[/attach]

where a(t) is the scale factor that describes the expansion of the universe, k is the curvature parameter that describes the spatial curvature of the universe (which can be positive, negative, or zero), and the other variables are as before.
How do these solutions suggest the existence of spacetime singularities, such as the curvature singularity at the center of a black hole, and what are the implications of these singularities for the physical observability and consistency of the theory?
One of the most striking features of these solutions is that they suggest the existence of spacetime singularities, where the curvature of spacetime becomes infinite. In the case of the Schwarzschild solution, the singularity is located at the center of the black hole, and is a curvature singularity. This means that certain components of the curvature tensor become infinite at the singularity. Similarly, in the Friedmann-Lemaître-Robertson-Walker metric, there is a singularity at the beginning of the universe, known as the Big Bang singularity. These singularities are problematic for the theory because they imply that the equations of general relativity break down at these points, and we do not have a complete theory of quantum gravity that can describe these extreme conditions.
Finally, how do the observations of the collapse of massive stars into black holes support the idea of spacetime singularities and the formation of black holes, and what are the theoretical and observational challenges in detecting and characterizing these singularities?
Observations of the collapse of massive stars into black holes provide strong evidence for the formation of spacetime singularities and the existence of black holes.

lightspeed2020 2023-4-20 16:59

A Hausdorff topological space M is a d-dimensional local Euclidean space if a point p in M has a neighborhood that is homeomorphic to an open set in R^d. A d-dimensional second-countable local Euclidean space with a differentiable structure is called a manifold.
On a smooth manifold, one can define a tangent space at any point. The basis vectors of this tangent space are denoted by e_µ, and the basis vectors of its dual space, the cotangent space (which maps a tangent vector to a linear function in R), are denoted by e^µ. General relativity (GR) is constructed on a manifold because it is coordinate-independent and enables calculus on curved surfaces.
From a physical perspective, a tangent vector (contravariant) at a point can be defined as a velocity vector, while a cotangent vector (covariant) represents the change in a specific direction at that point (e.g., the gradient). Flat spacetime is a special case of a manifold.
Manifolds are employed in physics to solve geometric problems, such as describing the trajectory of an object. Each point on a smooth manifold has a tangent space that describes its local properties, similar to a Euclidean space in the neighborhood of the point. Each tangent vector is linked to a curve, indicating the velocity and direction of the curve at that point.
Curvature measures the degree of curvature of a manifold, which can be defined in terms of geodesics on the surface and the radius of curvature for a given surface. In a flat space, the curvature is zero, indicating that all straight lines are parallel. In contrast, in a curved space, such lines are not parallel.
In general relativity, manifolds are used to describe spacetime, and curvature is used to describe the strength and distribution of the gravitational field. Black holes are an extreme example of curved spacetime, with curvature so significant that even light cannot escape their gravitational pull.

topochu 2023-4-20 22:42

lightspeed2020 2023-4-22 15:58

[attach]14209746[/attach]

topochu 2023-4-22 16:11

[attach]14209762[/attach]

topochu 2023-4-22 16:13

[attach]14209764[/attach]

[[i] 本帖最後由 topochu 於 2023-4-22 16:15 編輯 [/i]]

Goodman1945 2023-4-22 17:01

[quote]原帖由 [i]lightspeed2020[/i] 於 2023-4-22 15:58 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=557746619&ptid=31061597][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

[img]https://img.discuss.com.hk/d/attachments/day_230422/20230422_b1d1d6241be292b7757bAgHGQ0W21UCg.png[/img] [/quote]

lightspeed2020 2023-4-22 18:39

[quote]原帖由 [i]Goodman1945[/i] 於 2023-4-22 17:01 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=557747845&ptid=31061597][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]

These types of questions, such as quantum geometry and second-countable spaces, can easily be found online. While discussions are welcome, if the purpose of asking such questions is to test someone's knowledge, then it is unnecessary.
:loveliness:

topochu 2023-4-22 19:23

[[i] 本帖最後由 topochu 於 2023-4-22 19:32 編輯 [/i]]

lightspeed2020 2023-5-20 15:40

The suggestion is made that both the curvature and signature of spacetime may be subject to quantum fluctuations. This means that the structure of spacetime may be changing at a very small, quantum level.

In order to investigate this idea, a new metric for D-dimensional spacetime is introduced. This metric includes a phase factor, eiθ, which varies between 0 and π. At θ = 0, the metric describes Euclidean space, while at θ = π, it describes Minkowski space.

The behavior of this phase field is then considered, with the expectation value of the field determining the signature of spacetime. The effective potential for this field is calculated, and it is shown that in four dimensions, the minimum of the real part of the potential occurs at θ = π, which suggests a dynamical origin for the Lorentzian signature of spacetime.

lightspeed2020 2023-7-27 06:33

In quantum field theory, the meaning of Wick's theorem can be expressed using a physical formula. The statement of Wick's theorem is as follows:
Suppose we have a product of operators expressed as T(a₁ a₂ ... aₙ), where a₁, a₂, ..., aₙ are creation or annihilation operators.
Wick's theorem can be represented as:
T(a₁ a₂ ... aₙ) = :a₁ a₂ ... aₙ: + S₁ + S₂ + ... + Sₘ
where :a₁ a₂ ... aₙ: represents the normal-ordered product, which means arranging all creation operators to the left of annihilation operators in the correct order.
And S₁, S₂, ..., Sₘ represent all possible contractions, achieved by pairing creation and annihilation operators in the product and evaluating the expectation value between the paired operators. Each pairing generates a contraction term, resulting in m contraction terms in total.
Therefore, the physical formula expression of Wick's theorem is:
T(a₁ a₂ ... aₙ) = :a₁ a₂ ... aₙ: + S₁ + S₂ + ... + Sₘ
This formula illustrates how to decompose the product of operators into a normal-ordered product and a sum of contraction terms. The normal-ordered product signifies the correct ordering of the operators, while the contraction terms represent the pairings and expectation value calculations between operators. By using Wick's theorem, we can conveniently handle and compute physical quantities and interactions in quantum field theory.