[<< wikibooks] Introduction to Mathematical Physics/Relativity/Space geometrization
== Classical mechanics ==
Classical mechanics is based on two fundamental principles: the {\bf Galileo relativity} principle \index{Galileo relativity} and the fundamental principle of dynamics.
Let us state Galileo relativity principle:

In classical mechanics the time interval separating two events is
independent of the movement of the reference frame. Distance between two
points of a rigid body is independent of the movement of the
reference frame.

Following Gallilean relativity, the light speed should depend on the
Galilean reference frame considered. In 1881, the experiment of Michelson
and Morley attempting to measure this dependance fails.


== Relativistic mechanics (Special relativity) ==
Relativistic mechanics in the special case introduced by Einstein, as
he was 26 years old, is based on the following postulate: 

Because Einstein believes in the Maxwell equations (and because the
Michelson Morley experiment fails) 
  
    
      
        c
      
    
    {\displaystyle c}
   has to be a constant. So
Einstein postulates:

We will see how the physical laws have to be modified to obey to
those postulates later on\footnote{The fundamental laws of dyanmics
is deeply modified (see section secdynasperel (see section secdynasperel), but as guessed by Einstein
Maxwell laws obey to the special relativity postulates (see section seceqmaxcov.}.
The existence of a universal speed, the light speed, modifies deeply
space--time structure.
\index{space--time} It yields to precise the metrics\index{metrics} (see appendix chaptens for an introduction to the notion of
metrics) adopted in special relativity. Let us consider two Galilean reference frames characterized by coordinates: 
  
    
      
        (
        x
        ,
        t
        )
      
    
    {\displaystyle (x,t)}
   and 
  
    
      
        (
        
          x
          
            ′
          
        
        ,
        
          t
          
            ′
          
        
        )
      
    
    {\displaystyle (x^{\prime },t^{\prime })}
  . Assume that at 
  
    
      
        t
        =
        
          t
          
            ′
          
        
        =
        0
      
    
    {\displaystyle t=t^{\prime }=0}
   both
coordinate system coincide. Then:

that is to say:

and

Quantity 
  
    
      
        
          c
          
            2
          
        
        
          t
          
            2
          
        
        −
        
          x
          
            2
          
        
      
    
    {\displaystyle c^{2}t^{2}-x^{2}}
   is thus an invariant.
The most natural metrics that should equip space--time is thus:

It is postulated that this metrics should be invariant by Galilean change of
coordinates. 

Let us now look for the representation of a transformation of space--time that
keeps unchanged this metrics. We look for transformations such 
that:\index{Lorentz transformation}

is invariant.
From, the metrics, a "position vector" have to be defined. It is called
four-vector position, and two formalisms are possible to define it:\index{four--vector}.

Either coordinates of four-vector position are taken equal to 
  
    
      
        R
        =
        (
        x
        ,
        y
        ,
        z
        ,
        c
        t
        )
      
    
    {\displaystyle R=(x,y,z,ct)}
   and space is equipped by pseudo scalar product defined by matrix: 
  
    
      
        D
        =
        
          (
          
            
              
                
                  1
                
                
                  0
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
                
                  −
                  1
                
              
            
          
          )
        
      
    
    {\displaystyle D=\left({\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\\\end{array}}\right)}
  

 Then:  where 
  
    
      
        
          R
          
            t
          
        
      
    
    {\displaystyle R^{t}}
   represents the transposed of four-vector position 
  
    
      
        R
      
    
    {\displaystyle R}
  . 
Or coordinates of four-vector position are taken equal to 
  
    
      
        R
        =
        (
        x
        ,
        y
        ,
        z
        ,
        i
        c
        t
        )
      
    
    {\displaystyle R=(x,y,z,ict)}
   and space is equipped by pseudo scalar product defined by matrix:  
  
    
      
        D
        =
        I
        d
        =
        
          (
          
            
              
                
                  1
                
                
                  0
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
                
                  1
                
              
            
          
          )
        
      
    
    {\displaystyle D=Id=\left({\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{array}}\right)}
  

 Then:  where 
  
    
      
        
          R
          
            t
          
        
      
    
    {\displaystyle R^{t}}
   represents the transposed four-vector position 
  
    
      
        R
      
    
    {\displaystyle R}
  . 
Once the formalism is chosen, the representation of transformations ({\it i. e.,} the matrices), that leaves the pseudo-norm invariant can be investigated (see ([#References|references])). Here we will just exhibit such matrices. In first formalism, condition that pseudo-product scalar is invariant implies
that:

thus

Following matrix suits:

where 
  
    
      
        β
        =
        
          
            v
            c
          
        
      
    
    {\displaystyle \beta ={\frac {v}{c}}}
   (
  
    
      
        v
      
    
    {\displaystyle v}
   is the speed of the reference frame) and

  
    
      
        γ
        =
        
          
            1
            
              1
              −
              
                β
                
                  2
                
              
            
          
        
      
    
    {\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}}
  . 
The inverse of 
  
    
      
        M
      
    
    {\displaystyle M}
  :

In the second formalism, this same condition implies:

Following matrix suits:

and its inverse is:


=== Eigen time ===
Four-scalar (or Lorentz invariant) 
  
    
      
        d
        τ
      
    
    {\displaystyle d\tau }
   allows to define other
four-vectors (as four-vector velocity):

If mobile travels at velocity 
  
    
      
        v
      
    
    {\displaystyle v}
   in reference frame 
  
    
      
        
          R
          
            2
          
        
      
    
    {\displaystyle R_{2}}
  , then events A and B that are referenced in 
  
    
      
        
          R
          
            1
          
        
      
    
    {\displaystyle R_{1}}
   travelling the mobile by:

  
    
      
        
          
            
              
                
                  x
                  
                    A
                  
                
                =
                0
              
              
              
                
                  x
                  
                    B
                  
                
                =
                0
              
            
            
              
                c
                
                  t
                  
                    A
                  
                
                =
                0
              
              
              
                c
                
                  t
                  
                    B
                  
                
                =
                c
                τ
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}x_{A}=0&&x_{B}=0\\ct_{A}=0&&ct_{B}=c\tau \end{matrix}}}
  
and are referenced in 
  
    
      
        
          R
          
            2
          
        
      
    
    {\displaystyle R_{2}}
   by:

  
    
      
        
          
            
              
                
                  x
                  
                    A
                  
                
                =
                0
              
              
              
                
                  x
                  
                    B
                  
                
                =
                v
                t
              
            
            
              
                c
                
                  t
                  
                    A
                  
                
                =
                0
              
              
              
                c
                
                  t
                  
                    B
                  
                
                =
                c
                t
                .
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}x_{A}=0&&x_{B}=vt\\ct_{A}=0&&ct_{B}=ct.\end{matrix}}}
  
So, one gets the relation verified by 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  ::

so


=== Velocity four-vector ===
Velocity four-vector is defined by:

where 
  
    
      
        u
        =
        
          
            
              d
              x
            
            
              d
              t
            
          
        
      
    
    {\displaystyle u={\frac {dx}{dt}}}
   is the classical speed.


=== Other four-vectors ===
Here are some other four-vectors (expressed using first formalism):

four-vector position : 
  
    
      
        X
        =
        (
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        
          x
          
            3
          
        
        ,
        c
        t
        )
      
    
    {\displaystyle X=(x_{1},x_{2},x_{3},ct)}
  

 
four-vector wave: 
  
    
      
        K
        =
        (
        
          k
          
            1
          
        
        ,
        
          k
          
            2
          
        
        ,
        
          k
          
            3
          
        
        ,
        
          
            ω
            c
          
        
        )
      
    
    {\displaystyle K=(k_{1},k_{2},k_{3},{\frac {\omega }{c}})}
  

 
four-vector nabla: 
  
    
      
        K
        =
        (
        
          
            ∂
            
              ∂
              
                x
                
                  1
                
              
            
          
        
        ,
        
          
            ∂
            
              ∂
              
                x
                
                  2
                
              
            
          
        
        ,
        
          
            ∂
            
              ∂
              
                x
                
                  3
                
              
            
          
        
        ,
        
          
            ∂
            
              ∂
              
                x
                
                  4
                
              
            
          
        
        )
      
    
    {\displaystyle K=({\frac {\partial }{\partial x_{1}}},{\frac {\partial }{\partial x_{2}}},{\frac {\partial }{\partial x_{3}}},{\frac {\partial }{\partial x_{4}}})}
  


== General relativity ==
There exists two ways to tackle laws of Nature discovery problem: 

First method can be called {"phenomenological"}. A good example of phenomenological theory is quantum
mechanics theory. This method consists in starting from known facts (from
experiments) to infer laws. Observable notion is then a fundamental notion.

There exist another method less "anthropocentric" whose advantages had been underlined at century 17 by philosophers like Descartes.It is the method called {\it a priori}. It has been used by Einstein to propose his relativity theory. It consists in starting from principles that are believed to be true and to look for laws that obey to those principles.
Here are the fundamental postulates of general relativity:

Einstein believes strongly in those postulates. On another hand, he believes
that modelization of gravitational field have to be improved.
From this postulates, Einstein equation can be obtained: One can show that any
tensor 
  
    
      
        
          S
          
            i
            j
          
        
      
    
    {\displaystyle S_{ij}}
   that verifies those postulates:

where 
  
    
      
        a
      
    
    {\displaystyle a}
   and 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
   are two constants and 
  
    
      
        
          R
          
            i
            j
          
        
      
    
    {\displaystyle R_{ij}}
  , the Ricci
curvature tensor, and 
  
    
      
        R
      
    
    {\displaystyle R}
  , the scalar curvature are defined from 
  
    
      
        
          g
          
            i
            j
          
        
      
    
    {\displaystyle g_{ij}}
  
tensor\footnote{
Reader is invited to refer to specialized books for the expression of

  
    
      
        
          R
          
            i
            j
          
        
      
    
    {\displaystyle R_{ij}}
   and 
  
    
      
        R
      
    
    {\displaystyle R}
  .}
Einstein equation corresponds to 
  
    
      
        a
        =
        1
      
    
    {\displaystyle a=1}
  . Constant 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
   is called
cosmological constant. Matter tensor is not deduced from symmetries implied by postulates as tensor 
  
    
      
        
          S
          
            i
            j
          
        
      
    
    {\displaystyle S_{ij}}
   is. Please refer to [#References|references]) for indications about how to model matter tensor. Anyway, there is great difference between 
  
    
      
        
          S
          
            i
            j
          
        
      
    
    {\displaystyle S_{ij}}
   curvature tensor and matter tensor. Einstein opposes those two terms saying that curvature term is smooth as gold and matter term is rough as wood.