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*Orthogonal Transformation:*

Statement that laws of physics have same form in all inertial frames of reference is the expression of symmetry that is a characteristic property of nature. Entity is said to be symmetrical if after the operation on it, it remains same as it was before operation. The combination of transformation operations that leaves entity the same as it was before operations is known as a group. The example of group is rotation followed by reflection about axis. Other examples comprise Lorentz group which is translational motion at constant velocity followed by rotation and Poincaré group.

The transformation that leaves vector invariant after rotation of the axes of coordinate system is termed as orthogonal. Now, determine the specific geometrical transformation law for converting components of the vector from one frame to another.

Consider two frames of reference (coordinate systems) S and S' with the common origin Oas shown below:

S' is rotated through the angle φ in counter clockwise direction. Let θ and θ' be angles which position vector of a point P in a three-dimensional Euclidean space makes with x-axis and the x'-axis respectively. The common z-axis (not shown) is perpendicular to page and points toward you. From the figure we obtain

r_{x} = rcosθ, r_{y} = rsinθ

Also r'_{x} = rcosθ = rcos(θ - Φ) = r(cosθcosΦ + sinθsinΦ) = rxcosΦ + r_{y}sinΦ

Similarly

r'_{y} = rsinθ' = rsin(θ - Φ) = r(sinθcosΦ - cosθsinΦ) = -r_{x}sinΦ + r_{y}cosΦ

In addition, it can easily be seen that new coordinates (x, y, z) are linear combinations of old ones (x', y', z'). The coordinate transformation is said to be linear if coordinates are expressed as linear combination of old ones.

In matrix notation, our linear transformation of the coordinates of P can be written as

Usually, for the linear transformation of coordinates because of rotation about arbitrary axis in three dimensions, we can write:

x'_{1} = a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3}

x'_{2} = a_{21}x_{1} + a_{22}x_{2} + a_{23}x_{3}

x'_{3} = a_{31}x_{1} + a_{32}x_{2} + a_{33}x_{3}

And in matrix notation

We can write it as:

x'_{i} = Σ_{j=1}^{3}a_{ij}x_{j}

Coordinate transformation because of rotation from S' - frame relative to S-frame is indeed orthogonal. This will be true if length of the position vector of point P, that is |r|→, is invariant under coordinate transformation. Now, |r|→ in S frame is given as

|r^{→}|^{2} = x^{2} + y^{2} + z^{2}

|r'|^{→2 }= x'^{2} + y'^{2} + z'^{2}

= (xcosΦ + ysinΦ)2 + (xsinΦ + ycosΦ)^{2} + z^{2}

= x^{2} + y^{2} + z^{2}

Thus x'^{2 }+ y'^{2 }+ z'^{2 }= x^{2} + y^{2} + z^{2}

Thus, coordinate transformation is orthogonal under rotation.

In terms of Dirac delta notation, we write this as

Σ_{i}a_{ij}a_{ik }= δ_{jk}

where δ_{jk} = {0 if j ≠ k, 1 if j = k

Three significant points of this analysis are:

1. Components of other vectors transform in exactly the same way as displacement vector when axes of frames of reference are rotated with respect to each other.

2. New coordinates are mixtures of old ones. In other words, primed quantities are mixtures or combinations of unprimed ones. Length of the object in primed coordinate system is combination of length and width in unprimed system. As the example of this, if we set Φ = 90^{o}, then x' = y and y' = -x. This is the result of fact that we are capable to move around and look objects at different angles.

3. As transformation of kind we are dealing with is linear, sum of any two vectors will transform in same way when axes of reference frames are rotated relative to each other. For instance, Lorentz force given as F^{→} = q(E^{→} + v^{→} x B^{→})transforms in same way as . If equation is true for the set of axes, it is also true for axes at any other orientation.

*Lorentz Transformation as Orthogonal Transformation:*

Now apply analysis of orthogonal transformation of the vector in three dimensional Euclidean space to four dimensional space-time, the so-called Minkowski space. In relativity, space and time are intricately intertwined. Besides, Lorentz transformation is the linear combination of space and time coordinates. The event in space-time is thus completely specified by the vector in four-dimensional space with coordinates (x, y, z, ict). This kind of vector is known as a four-vector. The point in space-time is referred to as world point. Distance from origin to world point is known as interval.

The scalar product of three-dimensional vector (displacement vector) is invariant under rotational transformation. That is, r^{→}.r^{→} = |r^{→}|^{2} = x^{2} + y^{2} + z^{2} retains same form for all orientation of axes. Likewise, scalar product or square of length of four-vector s^{2} is invariant under Lorentz transformation. That is s^{2} = x^{2} + y^{2} + z^{2} - c^{2}t^{2} is Lorentz invariant. s^{2} will not always be positive compared with to corresponding r^{2} in three dimensions due to term -c^{2}t^{2}. So, in order that s^{2} might be analogous to r^{2} in three-dimensional vector, we have to replace t by ict in four-vector analysis, where ict should have dimensions of length.

*Algebra of Four-Vector:*

Now give formal definition of the four-vector. The four vector is, simply put, a vector in four dimensional real space known as Minkowski's space. By analogy to three-dimensional vector in Euclidean space, a four-vector can be expressed in terms of its components in four possible directions as the set of four quantities denoted by a_{μ} for μ = 1, 2, 3, 4 where 1, 2, 3 and 4 respectively refer to x, y, z and t. For example, momentum four-vector could be written in terms is components in four-dimensional space as P_{x}, P_{y}, P_{z}, P_{t} or simply P_{μ} where μ has values we have already stated and P_{t} is energy.

The coordinates of the event in four-dimensional space could be represented by x_{μ}.

Quantity a^{2}_{x} + a^{2}_{y} + a^{2}_{z} - a^{2}_{t} is invariant under complete Lorentz group. The four-vector transforms under complete Lorentz group, that is, for translational motion at constant velocity and rotation. Also, for two four-vector a_{μ} and b_{μ}, their corresponding components transform in same manner. So,

a_{x}b_{x} + a_{y}b_{y} + a_{z}b_{z} - a_{t}b_{t} = a^{→}.b^{→} - a_{t}b_{t} = a_{μ}b_{μ}

is invariant under complete Lorentz group, provided t is replaced by ict.

Therefore, we can write

a_{μ}b_{μ} = a'_{μ}b'_{μ}

The above is just an example of four-vector algebraic operation of scalar product

*Space-Time and its Geometry:*

**Space-Time:**

In the equations of Lorentz coordinate transformation, space and time coordinates are mixed. So, in relativity, we talk space-time as opposed to space only. Space-time is four dimensional space. The point in space-time is known as world point and defines event. Of course, you can reason that event takes place in space at some specified time. So, space-time is imaginary concept of space that has four dimensions that is x, y, z and t.

Distance between two world points is known as interval and is denoted by S_{12}, that is given as

S_{12 }= √[c^{2}(t_{2} - t_{1})^{2} - (x_{2} - x_{1})^{2} - (y_{2} - y_{1})^{2} - (z_{2} - z_{1})^{2}]

If interval is infinitesimally close, then we represent it as:

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

√c^{2}dτ^{2} - dl^{2}

Where, dl^{2} = dx^{2} + dy^{2} + dz^{2 }and τ = ict

The interval can be real or imaginary. For example, if t_{2} - t_{1} = 0, i.e., if event happen at time interval equal to zero, then interval is imaginary and is referred to as space-like interval. Condition to satisfy here is that

S_{12}^{2} < 0

For any space-like interval, there is inertial frame in which two events are simultaneous, but it is impossible to find the inertial frame in which two events happen at same place.

x_{2} = x_{1}, y_{2} = y_{1}, z_{2} = z_{1},

Then, two events occur at same point. This means that

S_{12}^{2} > 0

Interval is real and is referred to as time-like interval

**Space-Time Diagrams:**

In space-time continuum, plot space coordinates on horizontal axis while we plot time coordinate on vertical axis as shown in figure given below:

As you can see, vertical line represents the particle at rest. The particle travelling at speed of light (photon) is represented by the line inclined at angle of 45^{o} to positive x-axis. The particle going at ordinary speed (fraction of speed of light) like a rocket is represented by line whose slope

β = c/v

Path or trajectory of the particle in space-time continuum is known as world line.

Now, assume the moving observer (object, particle or signal) starts from origin x = 0 at t = 0. World line must be limited any point within light cone bounded by 45^{o} lines as no material object can travel at speed greater than that of light. Situation is illustrated in figure given below. Forward light cone represents observer's future as this is locus of all points available from start, origin that represents his present. Backward light cone represents observer's past. Somewhere else, that is, outside light cones, is unreachable to observer as he cannot travel faster than photon of light.

Implication of above is that inside the regions of light cone, signals can be sent at speeds less than speed of light from present to influence an event in future. That, is present can influence future. Likewise, past can influence present.

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