Lorentz Transformations: Videos & Practice Problems

Lorentz transformations are essential in special relativity for relating measurements between inertial frames, accounting for time dilation and length contraction. Unlike Galilean transformations, which simply add velocities, Lorentz transformations involve more complex equations. The x-component of velocity transforms as $v^{\text{'}}=(v-u)/(1-vu/c^2)$. Understanding these transformations is crucial for analyzing relativistic effects in high-speed scenarios.

Lorentz Transformations of Position

Lorentz transformations are essential in understanding how measurements of space and time change when observed from different inertial reference frames, particularly at speeds approaching the speed of light. Unlike Galilean transformations, which allow for simple addition of velocities, Lorentz transformations account for the effects of time dilation and length contraction, phenomena that arise from the principles of special relativity.

In Galilean relativity, the position in one reference frame can be easily related to another by the equation:

x' = x ± ut

where u is the relative velocity between the frames. However, as speeds approach the speed of light, this simple relationship becomes inadequate.

To derive the Lorentz transformations, we consider two inertial frames: the rest frame S and the moving frame S'. The origins of these frames are aligned at time t = t' = 0, and the relative velocity u is typically along the x-axis. The transformations can be expressed as follows:

x' = γ(x - ut)

t' = γ(t - (ux/c²))

Here, γ (the Lorentz factor) is defined as:

γ = \frac{1}{\sqrt{1 - \frac{u²}{c²}}}

where c is the speed of light. The Lorentz factor accounts for the relativistic effects that become significant at high velocities.

In the y and z directions, there is no relative motion, so the coordinates remain unchanged:

y' = y

z' = z

When applying these transformations, it is crucial to remember that time dilation occurs, meaning that time intervals measured in different frames will differ. The time dilation effect can be observed in the second term of the time transformation equation, which accounts for the relative motion between the frames.

For example, if we have a boost of 580 kilometers per second, we can calculate the Lorentz factor and subsequently determine the position and time in the moving frame S' using the above equations. The calculations will reveal that the effects of relativity are minimal at such speeds, as indicated by a Lorentz factor very close to 1.

In summary, Lorentz transformations provide a framework for understanding how measurements of time and space are interrelated in different inertial frames, particularly under relativistic conditions. They encapsulate the core principles of special relativity, allowing for accurate predictions of physical phenomena as velocities approach the speed of light.

In a lab frame, S, an object crosses a distance of 15 m in 10 s. In an initially aligned frame S', moving at 1000 km/s in the x-direction relative to S, how far a distance does the object have to travel, and in what time does it travel the distance?

• A. 0.99998889 s
• B. 9.9998889 s
• C. 1.0000056 s
• D. 10.000056 s

Answer: (D)

Lorentz Transformations of Velocity

In the study of special relativity, understanding Lorentz transformations is crucial, especially when analyzing velocities. Unlike Galilean relativity, where velocities can be simply added, relativistic velocities require a more complex approach due to phenomena such as length contraction and time dilation.

To apply Lorentz transformations for velocity, we consider two frames of reference, typically denoted as S and S'. The transformation for the x-component of velocity from frame S' (moving frame) to frame S (rest frame) is given by the equation:

$$v_x = \frac{v_x' - u}{1 - \frac{v_x' u}{c^2}}$$

Here, \(v_x'\) is the velocity in the moving frame, \(u\) is the speed of the moving frame relative to the rest frame, and \(c\) is the speed of light. This equation highlights that the transformation is not merely a straightforward addition of velocities, as it accounts for the relativistic effects that arise at high speeds.

For the y and z components of velocity, the transformations are slightly different. The y-component of velocity in the rest frame is given by:

$$v_y = \frac{v_y'}{\gamma \left(1 - \frac{v_x' u}{c^2}\right)}$$

And similarly for the z-component:

$$v_z = \frac{v_z'}{\gamma \left(1 - \frac{v_x' u}{c^2}\right)}$$

In these equations, \(v_y'\) and \(v_z'\) are the velocities in the moving frame, and \(\gamma\) (the Lorentz factor) is defined as:

$$\gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}}$$

This factor accounts for time dilation, which affects the measurement of time intervals in different frames. It is important to note that while the distances in the y and z directions remain unchanged, the time intervals do not, leading to a change in the observed velocities.

For example, consider a scenario where a spaceship is moving at 0.5c and fires a missile forward at 0.1c relative to the ship. To find the missile's speed as observed from Earth, we apply the transformation:

$$v_x = \frac{0.1c + 0.5c}{1 + \frac{0.1c \cdot 0.5c}{c^2}} = 0.57c$$

This result illustrates that the relativistic addition of velocities yields a speed less than the simple arithmetic sum, preventing any violation of the speed of light limit.

In another example, if a spaceship moves at 0.7c and fires a missile laterally at 0.2c, the transformation for the y-component must be applied. The calculation shows that the observed speed of the missile in the rest frame is:

$$v_y = \frac{0.2c}{\gamma}$$

With \(\gamma\) calculated based on the speed of the spaceship, this results in a reduced speed of 0.14c, demonstrating how velocities perpendicular to the direction of motion are also affected by relativistic effects.

In summary, Lorentz transformations are essential for accurately calculating velocities in special relativity, ensuring that the principles of relativity are upheld and that speeds do not exceed the speed of light.

In a particle accelerator, a neutron is traveling at a speed of 0.7 c, as measured by you in a laboratory. This neutron decays (becoming a proton), ejecting an electron. If you measure the electron's speed to be 0.5 c, traveling in the same direction as the neutron, what was the relative speed between the electron and neutron when the neutron decayed? Was the electron ejected forward or backwards relative to the neutron's motion, as 'seen' by the neutron?

• A. -0.55c; Forward
• B. -0.31c; Forward
• C. -0.31c; Backward
• D. -0.55c; Backward

Answer: (C)

Here’s what students ask on this topic:

What are Lorentz transformations and why are they important in special relativity?

Lorentz transformations are mathematical equations that relate the space and time coordinates of two inertial frames moving at a constant velocity relative to each other. They are crucial in special relativity because they account for the effects of time dilation and length contraction, which occur at high velocities close to the speed of light. Unlike Galilean transformations, which simply add velocities, Lorentz transformations ensure that the speed of light remains constant in all inertial frames, preserving the principles of special relativity. The transformations are given by:

$\frac{x\text{'}}{=\gamma (x-ut)}$

$\frac{t\text{'}}{=\gamma (t-\frac{ux}{c²})}$

where γ is the Lorentz factor, defined as:

$\gamma =\frac{1}{\sqrt{1-\frac{u²}{c²}}}$

How do Lorentz transformations differ from Galilean transformations?

Galilean transformations are used in classical mechanics to relate the coordinates of two inertial frames moving at a constant velocity relative to each other. They assume that time is absolute and the same for all observers, and they simply add velocities. The Galilean transformation for position is:

$x\text{'}=x-ut$

In contrast, Lorentz transformations are used in special relativity and take into account the constancy of the speed of light and the effects of time dilation and length contraction. They modify both space and time coordinates and are given by:

$\frac{x\text{'}}{=\gamma (x-ut)}$

$\frac{t\text{'}}{=\gamma (t-\frac{ux}{c²})}$

where γ is the Lorentz factor:

$\gamma =\frac{1}{\sqrt{1-\frac{u²}{c²}}}$

What is the Lorentz factor and how is it used in Lorentz transformations?

The Lorentz factor, denoted as γ (gamma), is a crucial component in Lorentz transformations. It accounts for the relativistic effects of time dilation and length contraction. The Lorentz factor is defined as:

$\gamma =\frac{1}{\sqrt{1-\frac{u²}{c²}}}$

where u is the relative velocity between the two inertial frames, and c is the speed of light. The Lorentz factor increases as the relative velocity approaches the speed of light, leading to more significant relativistic effects. In Lorentz transformations, γ is used to modify both space and time coordinates:

$\frac{x\text{'}}{=\gamma (x-ut)}$

$\frac{t\text{'}}{=\gamma (t-\frac{ux}{c²})}$

This ensures that the speed of light remains constant in all inertial frames, preserving the principles of special relativity.

How do Lorentz transformations affect time and length measurements?

Lorentz transformations significantly impact time and length measurements due to the effects of time dilation and length contraction. Time dilation means that a clock moving relative to an observer will tick slower compared to a clock at rest with respect to that observer. The time transformation is given by:

$\frac{t\text{'}}{=\gamma (t-\frac{ux}{c²})}$

Length contraction means that an object moving relative to an observer will appear shorter along the direction of motion. The length transformation is given by:

$\frac{x\text{'}}{=\gamma (x-ut)}$

where γ is the Lorentz factor:

$\gamma =\frac{1}{\sqrt{1-\frac{u²}{c²}}}$

These transformations ensure that the speed of light remains constant in all inertial frames, preserving the principles of special relativity.

How do you apply Lorentz transformations to velocity?

Applying Lorentz transformations to velocity involves more complex equations than for position. For the x-component of velocity, the transformation is:

$\frac{v\text{'}x}{=\frac{vx-u}{1-\frac{vxu}{c²}}}$

For the y and z components, the transformations are:

$\frac{v\text{'}y}{\frac{vy}{\gamma (1-\frac{vxu}{c²})}}$

$\frac{v\text{'}z}{\frac{vz}{\gamma (1-\frac{vxu}{c²})}}$

where γ is the Lorentz factor:

$\gamma =\frac{1}{\sqrt{1-\frac{u²}{c²}}}$

These transformations account for the relativistic effects of time dilation and ensure that the speed of light remains constant in all inertial frames.