General form of a Lorentz transformation

In summary, the Lorentz transformation is a mathematical transformation that converts the motion of one frame of reference relative to another. The general form of the transformation is x' = x + (γ-1)\frac{{\textbf{x}•\textbf{v}}}{v^2}\textbf{v}-γ\textbf{v}t' where x' represents the coordinate in the new frame of reference after the transformation, x represents the coordinate in the original frame of reference before the transformation, γ represents the constant of the transformation, t' represents the coordinate in the new frame of reference after the time has passed, and v represents the relative velocity of the two frames of reference. The
  • #1
rbwang1225
118
0
I got the general form of a Lorentz transformation in a GR book,

[itex]{\textbf{x'}}[/itex] = [itex]{\textbf{x}}[/itex] + (γ-1)[itex]\frac{{\textbf{x}•\textbf{v}}}{v^2}\textbf{v}[/itex]-γ[itex]\textbf{v}[/itex]t

t' = γ(t-[itex]\frac{{\textbf{x}•\textbf{v}}}{c^2}[/itex])

from frame S to frame S' moving with relative velocity [itex]\textbf{v}[/itex].

How to derive this?
 
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  • #2
The Lorentz transformation along the x-axis is

x' = γ(x - vt)
y' = y
z' = z
t' = γ(t - vx/c2)

This is

x' = γ(x - vt)
x' = x
t' = γ(t - 1/c2 v·x)

Put x = v v·x/v2 and x = x + x and you've got it.
 
  • #3
I don't understand what you mean. What is x = x + x?
 
  • #4
x represents the variable in the coordinate system for the spatial dimension that is oriented parallel to the vector pointing from the emitter to the detector (or detector to emitter)

x represents either of the variables in the other 2 spatial coordinates that are oriented perpendicular to the vector pointing from the emitter to the detector (or detector to emitter)

In other words, if x is parallel to the vector then x -> x while x -> y and z
Otherwise the relationship holds more generally
 
  • #5
There is another way of deriving this result. Suppose that the relative velocity of the S' frame of reference relative to the S frame of reference has components in all three coordinate directions. You can first rotate the cartesian spatial coordinates about the z axis to bring the velocity vector into a new x-z plane. Then you can apply a second rotation about the new y-axis to orient the relative velocity vector along a second new x axis. Then you can apply the boost in the second new coordinate system. After applying the boost, you can apply the inverse (= transpose) of the previous (combined) rotation matrices, and arrive at the transformed coordinates.

I like Bill K's approach better, because it is more elegant. He resolves an arbitrary position vector x into components perpendicular and parallel to the relative velocity vector. Only the component parallel to the velocity vector receives the boost. The component perpendicular to the velocity vector comes through unscathed.

Incidentally, this is not the most general form of the Lorentz Transformation. You can rigidly reorient the spatial coordinate axes of the S' frame of reference in any way you wish (by additional rotations) and still satisfy the condition that the relative velocity of the S frame of reference relative to the S' frame of reference has a magnitude equal to that of the S' frame of reference relative to the S frame of reference.
 

1. What is the general form of a Lorentz transformation?

The general form of a Lorentz transformation is a mathematical equation that describes the relationship between space and time coordinates in different reference frames. It is used to calculate how these coordinates change when an observer moves at a constant velocity relative to another observer.

2. What are the variables in the general form of a Lorentz transformation?

The variables in the general form of a Lorentz transformation are time (t), position in the x-direction (x), position in the y-direction (y), position in the z-direction (z), and the speed of light (c). These variables are used to calculate the transformation from one reference frame to another.

3. How is the general form of a Lorentz transformation derived?

The general form of a Lorentz transformation is derived from the principles of special relativity, which state that the laws of physics are the same in all inertial reference frames. It is derived using mathematical equations and concepts such as time dilation and length contraction.

4. What is the significance of the general form of a Lorentz transformation?

The general form of a Lorentz transformation is significant because it allows us to understand and calculate how time and space coordinates change between different reference frames. It is a fundamental concept in special relativity and is essential for understanding the effects of relative motion and the constancy of the speed of light.

5. What are some real-life applications of the general form of a Lorentz transformation?

The general form of a Lorentz transformation has many real-life applications, such as in the GPS system, which uses it to correct for time dilation effects caused by satellites moving at high speeds. It is also used in particle accelerators and other high-energy physics experiments to calculate the effects of relativistic speeds on subatomic particles.

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