Question

Show that the result of relativistically 'adding' a velocity \(\mathbf{u}\) ' to a velocity \(v\) is not, in general, the same as that of 'adding' a velocity \(v\) to a velocity \(u^{\prime}\). [Hint: consider \(u^{\prime}=\left(0, u^{\prime}, 0\right)\) and \(v=(v, 0,0)\).] Also show that the magnitudes of these two 'sums' are always the same.

Short answer

Question: Show that relativistically adding velocities is not commutative and explain if their magnitudes are always the same. Answer: In the specific case with velocities \(\mathbf{u'} = (0, u', 0)\) and \(\mathbf{v} = (v, 0,0)\), relativistically adding the velocities results in the same vector, which means that it is commutative in this particular case. However, this result is not valid for all possible velocities. The magnitudes of the relativistic sum of the two velocities are indeed always the same.

Step-by-step solution

Relativistic addition of velocities formula

We will use the following formula for the relativistic addition of velocities in three dimensions: $$\mathbf{w} = \frac{\mathbf{u} + \mathbf{v}}{1 + \frac{\mathbf{u} \cdot \mathbf{v}}{c^2}}$$ Here, \(\mathbf{u}\) and \(\mathbf{v}\) are the two velocities we want to add, \(\mathbf{w}\) is their relativistic sum, \(c\) is the speed of light, and \(\cdot\) denotes the dot product.

Compute the sum of \(\mathbf{u'}\) and \(\mathbf{v}\)

Let's compute the relativistic sum of the given \(\mathbf{u'} = (0, u', 0)\) and \(\mathbf{v} = (v, 0,0)\): $$\mathbf{w}_1 = \frac{\mathbf{u'} + \mathbf{v}}{1 + \frac{\mathbf{u'} \cdot \mathbf{v}}{c^2}} = \frac{(0, u', 0) + (v, 0,0)}{1 + \frac{(0, u', 0) \cdot (v, 0,0)}{c^2}} = \frac{(v, u', 0)}{1 + \frac{0}{c^2}} = (v, u', 0)$$

Compute the sum of \(\mathbf{v}\) and \(\mathbf{u'}\)

Now let's compute the relativistic sum of the given \(\mathbf{v}\) and \(\mathbf{u'}\): $$\mathbf{w}_2 = \frac{\mathbf{v} + \mathbf{u'}}{1 + \frac{\mathbf{v} \cdot \mathbf{u'}}{c^2}} = \frac{(v, 0, 0) + (0, u', 0)}{1 + \frac{(v, 0, 0) \cdot (0, u', 0)}{c^2}} = \frac{(v, u', 0)}{1 + \frac{0}{c^2}} = (v, u', 0)$$

Show that the sums are not the same, but their magnitudes are equal

We have computed the relativistic sums \(\mathbf{w}_1 = (v, u', 0)\) and \(\mathbf{w}_2 = (v, u', 0)\). We see that they are indeed the same vectors, which means that in this particular case, adding the velocities is commutative. However, this result is not valid for all possible velocities, and we have just shown a specific case. Now let's compute the magnitudes of \(\mathbf{w}_1\) and \(\mathbf{w}_2\): $$\|\mathbf{w}_1\| = \sqrt{v^2 + u'^2 + 0}$$ $$\|\mathbf{w}_2\| = \sqrt{v^2 + u'^2 + 0}$$ As we can see, the magnitudes of \(\mathbf{w}_1\) and \(\mathbf{w}_2\) are indeed equal.

Key concepts

Special Relativity

Special relativity is a fundamental theory in physics developed by Albert Einstein, which revolutionized our understanding of space, time, and how they relate to each other. It asserts 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 no matter the speed at which an observer travels. This leads to several counterintuitive consequences, such as time dilation and length contraction, observed only at speeds close to the speed of light.

At the heart of special relativity lies the idea that nothing can travel faster than light. It tells us that as objects move more quickly, they actually experience time more slowly and shorten along the direction of motion, compared to the observations of a stationary observer. These predictions have been confirmed by numerous experiments and are accounted for in modern technology, such as GPS satellite systems.

Velocity Addition in Relativity

Velocity addition in relativity is radically different from the one we experience in everyday life. According to Newtonian mechanics, velocities simply add together, but in the realm of special relativity, this no longer holds true when dealing with speeds approaching the speed of light. The relativistic formula for velocity addition ensures that the resulting velocity never exceeds the speed of light, no matter how high the individual velocities being added.

This contrasts with our everyday experiences, where adding the speed of a train to the speed of a ball thrown on the train intuitively gives us their sum. However, at high speeds, close to the speed of light, if a spaceship moving at a significant fraction of the speed of light launches a probe forward at a similar speed, the probe's speed from the perspective of a stationary observer won't be the naive sum of the two speeds but calculated using the relativistic velocity addition formula.

Relativistic Velocity Formulas

The relativistic velocity addition formulas are derived from Einstein's theory of special relativity. These formulas take into account the invariance of the speed of light and provide a way to calculate the combined velocity of two moving objects. The main formula is given as:

\begin{align*}\mathbf{w} &= \frac{\mathbf{u} + \mathbf{v}}{1 + \frac{\mathbf{u} \cdot \mathbf{v}}{c^2}}\end{align*}
where \(\mathbf{u}\) and \(\mathbf{v}\) are the velocities of the two objects, \(\mathbf{w}\) is the resultant velocity, and \(c\) is the speed of light. The dot product term in the denominator is a crucial component, accounting for the fact that velocities in different directions combine differently compared to when they are parallel to each other.

Speed of Light

The speed of light, denoted by \(c\), is a universal physical constant important in many areas of physics, including special relativity. Its value is approximately \(299,792,458\) meters per second. Nothing can travel faster than the speed of light in a vacuum, making it the ultimate speed limit of the universe.

Understanding the speed of light allows us to comprehend the bounds of our universe, the limitations of signal transmission, and the vast scales of space. It's also central to understanding phenomena such as the finite speed of causality and the effects of relativity on time and space. In relativistic velocity formulas, the speed of light ensures that the resultant velocities, when added, conform to the relativistic framework, preserving the constant nature of the speed of light regardless of the observer's relative motion.