### 物理代写|量子力学代写quantum mechanics代考|PHYS3040

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|量子力学代写quantum mechanics代考|Lorentzian and Galilean Spacetimes

For the sake of clarity, let us emphasise the meaning of lorentzian spacetime and galilean spacetime in intrinsic geometric terms (see $[22,104,129,260,309,310])$.
A lorentzian spacetime is defined to be a 4-dimensional affine space (special relativistic case), or a 4-dimensional manifold (general relativistic case), equipped with a lorentzian metric with signature $(-+++)$.

The signature of the lorentzian metric just selects the timelike and spacelike directions, but does not yield any preferred splitting of the lorentzian spacetime into space and time. Such a possible splitting requires (locally) the arbitrary choice of an observer.

A galilean spacetime is defined to be a 4-dimensional affine space (special relativistic case), or a 4-dimensional manifold (general relativistic case), equipped with a projection over absolute time and a galilean metric (spacelike euclidean metric, or spacelike riemannian metric) with signature $(0+++)$ (see Postulates C.1 and C.2).
The projection over absolute time selects the spacelike vector fields, but does not yield any preferred splitting of the galilean spacetime into space and time. Such a possible splitting requires (locally) the arbitrary choice of an observer. In order to get a preferred splitting into space and time, we would need an additional preferred projection over space.

Thus, an essential comparison between the galilean spacetime and the einsteinian spacetime can be summarised as follows: in the 1st case we have a time fibring and a spacelike riemannian metric, in the 2nd case the time fibring is missing and the spacelike riemannian metric is replaced by a spacetime lorentzian metric.

## 物理代写|量子力学代写quantum mechanics代考|Principle of Relativity

In the standard physical literature, for clear historical reasons, the words “covariance”, “covariant”, “relativity” and “relativistic” are largely used in strict connection with einsteinian Special and General Relativity. However, the above standard usage of these words might be quite misleading in the context of the present book. So,here we establish, without any pretension of completeness and full rigour, linguistic conventions which are suitable for our discussion.

Going back to the original Einstein’s work, we might say, in a few words, that a relativistic theory is defined to be a physical theory whose fundamental laws can be expressed in an observer equivariant way. Such a condition requires to state which are the admissible observers of the theory we are dealing with. So, in Special and General Relativity the fundamental physical laws are, respectively, equivariant with respect to inertial and general observers.

Actually, in the Einstein theory, spacetime is a lorentzian affine space (Minkowski space of Special Relativity) or a lorentzian manifold (spacetime of General Relativity). Accordingly, the selection of distinguished observers (inertial or general observers) depends on the background lorentzian structure of spacetime. Therefore, in the Einstein theory, there is an essential interplay of the lorentzian structure of spacetime and the principle of relativity.

With reference to a generic physical theory, the principle of relativity, understood as equivariance of fundamental physical laws with respect to observers, can be detached from the possible lorentzian structure of spacetime.

For instance, we may formulate a theory of flat galilean spacetime in an equivariant way with respect to inertial observers. Indeed, such a formulation can also be extended to a curved galilean spacetime and to general observers. By keeping the above general meaning of relativistic theory, we might say that such galilean theories are relativistic.

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