Introduction to Mathematical Physics/Relativity/Space geometrization

Classical mechanics

Classical mechanics is based on two fundamental principles: the Galilean relativity principle and the fundamental principle of dynamics. Let us state Galileo relativity principle:

Principle: Galileo relativity principle. Classical mechanics laws (in particular Newton's law of motion) have the same form in every frame in uniform translation with respect to each other. Such frames are called Galilean frames or inertial frames.

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.

Remark: Classical mechanics laws are invariant by transformations belonging to Galileo transformation group. A Galileo transformation of coordinates can be written:

${\begin{matrix}x'_{1}&=&x_{1}-vt\\x'_{1}&=&x_{1}\\x'_{1}&=&x_{1}\\t'&=&t\end{matrix}}$

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.

secrelat

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:

Postulate: All the laws of Universe i. e. laws of mechanics and electromagnetism) are the same in all Galilean reference frames.

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

Postulate: The light speed in vacuum $c$ is the same in every Galilean reference frame. This speed is an upper bound.

We will see how the physical laws have to be modified to obey to those postulates later on . The existence of a universal speed, the light speed, modifies deeply space--time structure. It yields to precise the 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: $(x,t)$ and $(x^{\prime },t^{\prime })$. Assume that at $t=t^{\prime }=0$ both coordinate system coincide. Then:

$c={\frac {|x|}{|t|}}={\frac {|x^{\prime }|}{|t^{\prime }|}}$

that is to say:

$c^{2}t^{2}-x^{2}=0$

and

$c^{2}t^{\prime 2}-x^{\prime 2}=0$

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

$ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}$

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

Postulate: Metrics $ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}$ is invariant by change of Galilean reference frame.

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

$ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}$

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

Once the formalism is chosen, the representation of transformations i. e., the matrices), that leaves the pseudo-norm invariant can be investigated (see (references)). Here we will just exhibit such matrices. In first formalism, condition that pseudo-product scalar is invariant implies that:

$(MR|MR)=(R|R)$

thus

cond

$D=M^{+}DM$

Following matrix suits:

$M=\left({\begin{array}{cccc}\gamma &0&0&\gamma \beta \\0&1&0&0\\0&0&1&0\\\gamma \beta &0&0&\gamma \\\end{array}}\right)$

where $\beta ={\frac {v}{c}}$ ($v$ is the speed of the reference frame) and $\gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}$. The inverse of $M$:

$M^{-1}=\left({\begin{array}{cccc}\gamma &0&0&-\gamma \beta \\0&1&0&0\\0&0&1&0\\-\gamma \beta &0&0&\gamma \\\end{array}}\right)$

Remark:

Equation cond implies a condition for the determinant:

$1=det(M^{+}DM)=(det(M))^{2}$

Matrices $M$ of determinant 1 form a group called Lorentz group. \index{Lorentz group}

In the second formalism, this same condition implies:

$Id=M^{+}M$

Following matrix suits:

$M=\left({\begin{array}{cccc}\gamma &0&0&-i\gamma \beta \\0&1&0&0\\0&0&1&0\\+i\gamma \beta &0&0&\gamma \\\end{array}}\right)$

and its inverse is:

$M^{-1}=\left({\begin{array}{cccc}\gamma &0&0&+i\gamma \beta \\0&1&0&0\\0&0&1&0\\-i\gamma \beta &0&0&\gamma \\\end{array}}\right)$

Remark: A unitary matrix (see section secautresrep) is a matrix such that:

$M^{+}M=1=MM^{+}$

where $M^{+}$ is the adjoint matrix of $M$, that is the conjugated transposed matrix of $M$. Then, scalar product defined by:

${\mathrel {<}}R|Q{\mathrel {>}}=R^{+}Q$

is preserved by the action of $M$.

Eigen time

Four-scalar (or Lorentz invariant) $d\tau$ allows to define other four-vectors (as four-vector velocity):

Definition: Eigen time of a mobile is time marked by a clock travelling with this mobile.

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

${\begin{matrix}x_{A}=0&&x_{B}=0\\ct_{A}=0&&ct_{B}=c\tau \end{matrix}}$

and are referenced in $R_{2}$ by:

${\begin{matrix}x_{A}=0&&x_{B}=vt\\ct_{A}=0&&ct_{B}=ct.\end{matrix}}$

So, one gets the relation verified by $\tau$::

$\tau ^{2}=t^{2}(1-{\frac {v^{2}}{c^{2}}})$

so

$d\tau ={\frac {dt}{\gamma }}$

Velocity four-vector

Velocity four-vector is defined by:

$U={\frac {dX}{d\tau }}=(\gamma u,\gamma c)$

where $u={\frac {dx}{dt}}$ is the classical speed.

Other four-vectors

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

• four-vector position :
  $X=(x_{1},x_{2},x_{3},ct)$
• four-vector wave:
  $K=(k_{1},k_{2},k_{3},{\frac {\omega }{c}})$
• four-vector nabla:
  $K=({\frac {\partial }{\partial x_{1}}},{\frac {\partial }{\partial x_{2}}},{\frac {\partial }{\partial x_{3}}},{\frac {\partial }{\partial x_{4}}})$

General relativity

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

1. 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.

1. There exist another method less "anthropocentric" whose advantages had been underlined at century 17 by philosophers like Descartes.

It is the method called 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:

Postulate: Generalized relativity principle: All the laws of Nature are covariant\index{covariance} relatively to any continuous transformation of coordinates system. <ref>Special relativity states only covariance with respect to Lorentz transformations (see (references)) and classical mechanics states covariance only under Galilean transformations.<\ref>

Postulate: Maximum logical simplicity principle for laws formulation: All geometrical properties of space--time can be described by the means of a differential tensor $S$. This tensor

1. is expressed in a four dimension Riemannian space whose metrics is defined by a tensor $g_{ij}$
2. is a second order tensor and is noted $S_{ij}$
3. is a function of the $g_{ij}$'s that doesn't contain any partial derivatives of order greater than two and that is linear with respect to second order partial derivatives.

Postulate: Divergence of tensor $S_{ij}$ is zero.

Postulate: Space curvature is due to matter:

${\mbox{ Curvature }}={\mbox{ Matter }}$

or, using tensors:

$S_{ij}=T_{ij}$

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 $S_{ij}$ that verifies those postulates:

$S_{ij}=a(R_{ij}-{\frac {1}{2}}g_{ij}R-\lambda g_{ij})$

where $a$ and $\lambda$ are two constants and $R_{ij}$, the Ricci curvature tensor, and $R$, the scalar curvature are defined from $g_{ij}$ tensor\footnote{ Reader is invited to refer to specialized books for the expression of $R_{ij}$ and $R$.} Einstein equation corresponds to $a=1$. Constant $\lambda$ is called cosmological constant. Matter tensor is not deduced from symmetries implied by postulates as tensor $S_{ij}$ is. Please refer to references) for indications about how to model matter tensor. Anyway, there is great difference between $S_{ij}$ 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.