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Lorentz transformation

In physics, the Lorentz transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicistHendrik Lorentz. It reflects the surprising fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events.

The Lorentz transformation was originally the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. Albert Einstein later re-derived the transformation from his postulates of special relativity. The Lorentz transformation supersedes the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). According to special relativity, this is a good approximation only at relative speeds much smaller than the speed of light.

If space is homogeneous, then the Lorentz transformation must be a linear transformation. Also, since relativity postulates that the speed of light is the same for all observers, it must preserve the spacetime interval between any two events in Minkowski space. The Lorentz transformation describes only the transformations in which the spacetime event at the origin is left fixed, so they can be considered as a rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.

Lorentz transformation for frames in standard configuration

Assume there are two observers O and Q, each using their own Cartesian coordinate system to measure space and time intervals. O uses (t, x, y, z) and Q uses (t', x', y', z'). Assume further that the coordinate systems are oriented so that the x-axis and the x' -axis are collinear, the y-axis is parallel to the y' -axis, as are the z-axis and the z' -axis. The relative velocity between the two observers is v along the common x-axis. Also assume that the origins of both coordinate systems are the same. If all these hold, then the coordinate systems are said to be in standard configuration. A symmetric presentation between the forward Lorentz Transformation and the inverse Lorentz Transformation can be achieved if coordinate systems are in symmetric configuration. The symmetric form highlights that all physical laws should be of such a kind that they remain unchanged under a Lorentz transformation.

The Lorentz transformation for frames in standard configuration can be shown to be:


t' &= \gamma \left( t - v x/c^{2} \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end{cases} where \ \gamma = \frac{1}{ \sqrt{1 - { \frac{v^2}{c^2}}}} is called the Lorentz factor.

Matrix form

This Lorentz transformation is called a "boost" in the x-direction and is often expressed in matrix form as

\begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix}

\begin{bmatrix} \gamma&-\beta \gamma&0&0\\ -\beta \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}\ . More generally for a boost in any arbitrary direction (\beta_{x}, \beta_{y}, \beta_{z}),

\begin{bmatrix} c\,t' \\ x' \\ y' \\ z' \end{bmatrix}

\begin{bmatrix} \gamma&-\beta_x\,\gamma&-\beta_y\,\gamma&-\beta_z\,\gamma\\ -\beta_x\,\gamma&1+(\gamma-1)\dfrac{\beta_{x}^{2}}{\beta^{2}}&(\gamma-1)\dfrac{\beta_{x}\beta_{y}}{\beta^{2}}&(\gamma-1)\dfrac{\beta_{x}\beta_{z}}{\beta^{2}}\\ -\beta_y\,\gamma&(\gamma-1)\dfrac{\beta_{y}\beta_{x}}{\beta^{2}}&1+(\gamma-1)\dfrac{\beta_{y}^{2}}{\beta^{2}}&(\gamma-1)\dfrac{\beta_{y}\beta_{z}}{\beta^{2}}\\ -\beta_z\,\gamma&(\gamma-1)\dfrac{\beta_{z}\beta_{x}}{\beta^{2}}&(\gamma-1)\dfrac{\beta_{z}\beta_{y}}{\beta^{2}}&1+(\gamma-1)\dfrac{\beta_{z}^{2}}{\beta^{2}}\\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}\ , where \beta = \frac{v}{c}=\frac{c} and \gamma = \frac{1}{\sqrt{1-\beta^2}}.

Note that this transformation is only the "boost," i.e., a transformation between two frames whose x, y , and z axis are parallel and whose spacetime origins coincide (see The "Standard configuration" Figure). The most general proper Lorentz transformation also contains a rotation of the three axes, because the composition of two boosts is not a pure boost but is a boost followed by a rotation. The rotation gives rise to Thomas precession. The boost is given by a symmetric matrix, but the general Lorentz transformation matrix need not be symmetric.

The composition of two Lorentz boosts B(u) and B(v) of velocities u and v is given by:


where u\oplusv is the velocity-addition, and Gyr[u,v] is the rotation arising from the composition, gyr being the gyrovector space abstraction of the gyroscopic Thomas precession, and B(v) is the 4x4 matrix that uses the components of v, i.e. v1, v2, v3 in the entries of the matrix, or rather the components of v/c in the representation that is used above.


From Yahoo Answers

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Question:can you people out there give me all the formulas that you know about physics?especially those important ones..(needed for the 4th year high school, Philippines Curriculum) by the way, thnks!

Answers:*One dimensional motion There are four variables which put together in an equation can describe this motion. These are Initial Velocity (u); Final Velocity (v), Acceleration (a), Distance Traveled (s) and Time elapsed (t). The equations which tell us the relationship between these variables are as given below. v = u + at v2 = u2 + 2 as click for calculator s = ut + 1/2 at2 average velocity = (v + u)/2 *Newton's laws of motion Through Newton's second law, which states: The acceleration of a body is directly proportional to the net unbalanced force and inversely proportional to the body's mass, a relationship is established between Force (F), Mass (m) and acceleration (a). This is of course a wonderful relation and of immense usefulness. F = m x a *Momentum (p) is the quantity of motion in a body. A heavy body moving at a fast velocity is difficult to stop. A light body at a slow speed, on the other hand can be stopped easily. So momentum has to do with both mass and velocity. p = mv *Impulse is the change in the momentum of a body caused over a very short time. Let m be the mass and v and u the final and initial velocities of a body. Impulse = Ft = mv - mu ===================== Work and energy As we know from the law of conservation of energy: energy is always conserved. *Work is the product of force and the distance over which it moves. Imagine you are pushing a heavy box across the room. The further you move the more work you do! If W is work, F the force and x the distance then. W = Fx *Energy comes in many shapes. The ones we see over here are kinetic energy (KE) and potential energy (PE) Transitional KE = mv2 Rotational KE = Iw2 here I is the moment of inertia of the object (a simple manner in which one can understand moment of inertia is to consider it to be similar to mass in transitional KE) a w is angular velocity Gravitational PE = mgh where h is the height of the object Elastic PE = k L 2 where k is the spring constant ( it gives how much a spring will stretch for a unit force) and L is the length of the spring. *Power Power (P) is work( W) done in unit time (t). P = W/t as work and energy (E) are same it follows power is also energy consumed or generated per unit time. P = E/t In measuring power Horsepower is a unit which is in common use. However in physics we use Watt. So the first thing to do in solving any problem related to power is to convert horsepower to Watts. 1 horsepower (hp) = 746 Watts *Circular motion a = v2 / r F = ma = mv2/r *Newton's law of universal gravitation About fifty years after Kepler announced the laws now named after him, Isaac Newton showed that every particle in the Universe attracts every other with a force which is proportional to the products of their masses and inversely proportional to the square of their separation. Hence: If F is the force due to gravity, g the acceleration due to gravity, G the Universal Gravitational Constant (6.67x10-11 N.m2/kg2), m the mass and r the distance between two objects. Then F = G m1 m2 / r2 *Acceleration due to gravity outside the Earth It can be shown that the acceleration due to gravity outside of a spherical shell of uniform density is the same as it would be if the entire mass of the shell were to be concentrated at its center. Using this we can express the acceleration due to gravity (g') at a radius (r) outside the earth in terms of the Earth's radius (re) and the acceleration due to gravity at the Earth's surface (g) g' = (re2 / r2) g *Acceleration due to gravity inside the Earth Here let r represent the radius of the point inside the earth. The formula for finding out the acceleration due to gravity at this point becomes: g' = ( r / re )g In both the above formulas, as expected, g' becomes equal to g when r = re. *Density The mass of a substance contained in unit volume is its density (D). D = m/V Measuring of densities of substances is easier if we compare them with the density of some other substance of know density. Water is used for this purpose. The ratio of the density of the substance to that of water is called the Specific Gravity (SG) of the substance. SG = Dsubstance / Dwater The density of water is 1000 kg/m3 *Pressure Pressure (P) is Force (F) per unit area (A) P = F/A Electricity According to Ohm's Law electric potential difference(V) is directly proportional to the product of the current(I) times the resistance(R). V = I R The relationship between power (P) and current and voltage is P = I V Using the equations above we can also write P = V2 / R and P = I2 R

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