In the rest frame, the coordinates of the ends of the rod in...

In the rest frame, the coordinates of the ends of the rod in terms of proper length $l_{0}$ $$ A:(0,0,0) B:\left(l_{0} \cos \theta_{0}, l_{0} \sin \theta_{0}, 0\right) $$ at time $t$. In the laboratory frame the coordinates at time $t^{\prime}$ are $$ A:\left(v t^{\prime}, 0,0\right), B:\left(l_{0} \cos \theta_{0} \sqrt{1-\beta^{2}}+v t^{\prime}, l_{0} \sin \theta_{0}, 0\right) $$ $\mathbf{1 6 9}$ Therefore we can write, $l \cos \theta_{0}=l_{0} \cos \theta_{0} \sqrt{1-\beta^{2}}$ and $l \sin \theta=l_{0} \sin \theta_{0}$ Hence $l_{0}^{2}=\left(l^{2}\right)\left(\frac{\cos ^{2} \theta+\left(1-\beta^{2}\right) \sin ^{2} \theta}{1-\beta^{2}}\right)$ or, $=\sqrt{\frac{1-\beta^{2} \sin ^{2} \theta}{1-\beta^{2}}}$

In the rest frame, the coordinates of the ends of the rod in terms of proper length $l_{0}$
$$
A:(0,0,0) B:\left(l_{0} \cos \theta_{0}, l_{0} \sin \theta_{0}, 0\right)
$$
at time $t$. In the laboratory frame the coordinates at time $t^{\prime}$ are
$$
A:\left(v t^{\prime}, 0,0\right), B:\left(l_{0} \cos \theta_{0} \sqrt{1-\beta^{2}}+v t^{\prime}, l_{0} \sin \theta_{0}, 0\right)
$$
$\mathbf{1 6 9}$
Therefore we can write, $l \cos \theta_{0}=l_{0} \cos \theta_{0} \sqrt{1-\beta^{2}}$ and $l \sin \theta=l_{0} \sin \theta_{0}$
Hence $l_{0}^{2}=\left(l^{2}\right)\left(\frac{\cos ^{2} \theta+\left(1-\beta^{2}\right) \sin ^{2} \theta}{1-\beta^{2}}\right)$
or, $=\sqrt{\frac{1-\beta^{2} \sin ^{2} \theta}{1-\beta^{2}}}$

Key Concepts

Coordinate Transformation

Coordinate transformation involves applying the Lorentz transformation equations to convert the space-time coordinates of events from one inertial frame to another. This process is key to understanding how quantities like position, length, and time intervals change due to relative motion between observers.

Relativity of Simultaneity

The relativity of simultaneity is the principle that events perceived as simultaneous in one inertial frame may not be simultaneous in another moving frame. This concept is crucial in the measurement of spatial distances, such as the length of a moving object, because these measurements depend on the time at which the endpoints are recorded in a given frame.

Frames of Reference

Frames of reference are coordinate systems used to measure and describe physical phenomena. In relativity, different observers may be in different inertial frames – for example, the rest frame of an object versus the laboratory frame – leading to differing measurements of space and time coordinates for the same events.

Lorentz Contraction

Lorentz contraction refers to the phenomenon in special relativity where an object moving relative to an observer appears shorter along the direction of its motion than when measured in its rest frame. This contraction is quantified by the Lorentz factor, which depends on the relative velocity between the observer and the object.

Proper Length

Proper length is the length of an object measured in the frame where the object is at rest. It serves as an invariant reference value in special relativity from which the effects of relative motion, such as length contraction, are computed.

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