Question

Observer \(S\) assigns to an event the coordinates \(x=100 \mathrm{~km}\), \(t=200 \mu \mathrm{s}\). Find the coordinates of this event in frame \(S^{\prime}\), which moves in the direction of increasing \(x\) with speed \(0.950 c\). Assume that \(x=x^{\prime}\) at \(t=t^{\prime}=0\).

Short answer

The coordinates of the event in frame \(S^{\prime}\) would be: \(x^{\prime} \approx 9.6 × 10^4 m\) and \(t^{\prime} \approx 1.8 × 10^−4 s\).

Step-by-step solution

Convert Units

First, convert all units into fundamental SI units for consistency. The given distance is in kilometres and the time in microseconds, so they both need to be converted into meters and seconds respectively. So, \(x = 100km = 100 \times 10^3 m = 1.0 \times 10^5 m\), and \(t = 200 \mu s = 200 \times 10^{-6} s = 2.0 \times 10^{-4} s\).

Use Lorentz Transformations

The Lorentz transformation equations for position and time can be expressed as follows: \(x^{\prime} = \gamma (x- vt)\), \(t^{\prime} = \gamma (t - \frac{vx}{c^2})\) where, \(v\) is the velocity of \(S^{\prime}\) relative to \(S\), \(c\) is the speed of light, and \(\gamma\) is the Lorentz factor defined as \(\frac{1}{\sqrt{1- (\frac{v}{c})^2}}\). Knowing that \(v = 0.950c\), we can first calculate \(\gamma\).

Calculate \(\gamma\)

First of all we calculate the Lorentz factor. \(\gamma = \frac{1}{\sqrt{1- (\frac{v}{c})^2}} = \frac{1}{\sqrt{1- (0.950)^2}} \approx 3.20\)

Calculate \(x^{\prime}\) and \(t^{\prime}\)

Now, let's calculate \(x^{\prime}\) and \(t^{\prime}\) using the Lorentz transformation equations and the values we have obtained so far. The Lorentz transformation for the position gives: \(x^{\prime} = \gamma (x - vt) \approx 3.20(1.0 \times 10^5 m - 0.950c \times 2.0 \times 10^{-4}s) \approx 9.6 × 10^4 m\). Similarly, the Lorentz transformation for the time gives: \(t^{\prime} = \gamma (t - \frac{vx}{c^2}) \approx 3.20 (2.0 \times 10^{-4}s - \frac{0.950 \times 1.0 \times 10^5 m}{c^2}) \approx 1.8 × 10^−4 s\).

Key concepts

Special Relativity

Special relativity is a theory proposed by Albert Einstein that revolutionizes the understanding of space and time. The central idea is that the laws of physics are the same for all non-accelerating observers, and that the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer. This leads to fascinating phenomena such as time dilation and length contraction, which are not noticeable in everyday life but become significant when objects move at a high fraction of the speed of light.

Incorporating the exercise from our example, special relativity is crucial when calculating the coordinates of an event as seen by observers moving relative to each other at a high velocity. The Lorentz transformation, which is derived from this theory, provides the mathematical tool needed to convert coordinates from one frame to another.

Reference Frames

Reference frames are essentially viewpoints from which measurements such as position, velocity, and time are made. In physics, 'inertial reference frames' are ones where objects not acted upon by forces move at a constant velocity. It's like watching the world from a smoothly gliding train — unless you look outside, you can't tell you're moving. Conversely, objects may seem to behave differently when observed from 'non-inertial reference frames', which accelerate or rotate.

Considering our exercise, an observer in frame S and another in a moving frame S' might disagree on the coordinates of an event because they are using different reference frames. Transforming coordinates, like using the Lorentz transformation, is key to understanding how different observers perceive the same event.

Time Dilation

Time dilation is a direct consequence of the postulates of special relativity. It refers to the effect that a clock moving relative to an observer will be measured to tick slower than a clock that is at rest with respect to the observer. This effect becomes increasingly significant as the relative speed approaches the speed of light. Put simply, time runs at different rates for people in different reference frames moving relative to one another.

In our exercise, time dilation is encapsulated in the Lorentz transformation for time t'. The observer moving at a speed of 0.950c compared to another will measure the time interval of an event differently due to this astounding effect.

Length Contraction

Length contraction is another peculiar consequence of special relativity, stating that objects contract along the direction of motion as their speed approaches the speed of light. So, if you were to observe a spaceship flying by at near-light speeds, it would appear shorter to you. However, to the people on the spaceship, the lengths would seem normal. This effect, like time dilation, becomes only noticeable at speeds close to the speed of light.

In the context of our problem, length contraction is not directly assessed, but is inherently part of the Lorentz transformations applied. It's crucial for understanding the observed shortening of measurements in the event's position on the moving frame S'.

Speed of Light

The speed of light in a vacuum, commonly represented by the symbol c, is one of the fundamental constants of nature, with a value of approximately 299,792,458 meters per second. According to special relativity, this speed is the ultimate speed limit of the universe and remains constant for all observers, regardless of their motion relative to the light source. Not only does c feature in calculations of electromagnetic phenomena, but it is also intrinsically tied to the structure of spacetime as shown in the Lorentz transformations used in our problem-solving exercise.

The speed at which frame S' is moving relative to S is expressed as a fraction of c, which signifies just how fast that is in human terms and why relativistic effects must be accounted for in calculating the changed coordinates of an event.