Question

Galaxy A is reported to be receding from us with a speed of \(0.347 c\). Galaxy B, located in precisely the opposite direction, is also found to be receding from us at this same speed. What recessional speed would an observer on galaxy A find \((a)\) for our galaxy and \((b)\) for galaxy \(\mathrm{B}\) ?

Short answer

The observer on Galaxy A would find a speed of \(-0.347c\) for our galaxy, and a speed of \(0.347c\) for Galaxy B.

Step-by-step solution

Understanding the Problem

Galaxy A and Galaxy B are moving in opposite directions with a speed of \(0.347c\) each. We need to find the speed of both galaxies from the perspective of an observer on Galaxy A.

Application of Relativistic Velocity Addition

Using the formula for the relativistic addition of velocities, \(v_{ab}=\frac{v_a+v_b}{1+\frac{v_a v_b}{c^2}}\), where \(v_a\) and \(v_b\) are the velocities of galaxies A and B respectively, and \(v_{ab}\) is the velocity of galaxy B as observed from galaxy A.

Calculation of Speed of Our Galaxy as Observed from Galaxy A (part a)

The observer on Galaxy A will see our galaxy ('us') moving away. Therefore, \(v_a\) (or our speed) is \(-0.347c\), while \(v_b\) is \(0\), as Galaxy A is stationary according to the observer on it. Using the relativistic addition of velocities formula, we get \(v_{ab}= -0.347c\).

Calculation of Speed of Galaxy B as Observed from Galaxy A (part b)

The observer on Galaxy A will see Galaxy B moving away. This time, we keep \(v_a=0\) (as the observer's galaxy, Galaxy A, is stationary to him) and \(v_b=0.347c\) (the speed of Galaxy B). Using the same formula, we find that \(v_{ab}=0.347c\) or in other words, an observer from Galaxy A will see Galaxy B moving away at a speed of \(0.347c\).

Key concepts

Special Relativity

Special relativity is a fundamental theory in the field of physics that Albert Einstein introduced in 1905. It revolutionized our understanding of space, time, and how they interact with each other and with matter at high velocities. One core principle is that the speed of light in a vacuum (\( c \) approximately 299,792,458 meters per second) is constant and is not dependent on the observer's frame of reference.

This means that no matter how fast you are moving, the speed of light remains the same, even if you are traveling in the direction of the light beam or away from it. This leads to various counterintuitive implications, such as time dilation (moving clocks run slower) and length contraction (moving objects shorten).

However, it's important to note that these effects only become significant at speeds close to the speed of light (\( c \)). For everyday velocities, the Newtonian mechanics rules, which assume absolute time and space, are sufficiently precise. But for astronomical objects moving at a significant fraction of the speed of light, such as galaxies, we must use special relativity to describe their behavior accurately.

Velocity of Galaxies

When we talk about the 'velocity of galaxies,' we're referring to their speed relative to some point of reference, often another galaxy or a point in space. Galaxies can have different kinds of motion, like rotational around their own axes and orbital around other bodies, but one key aspect is their motion relative to each other due to expansion of the universe.

Because space itself is expanding, galaxies are carried along with that expansion, which means they typically recede from each other, a motion that can be measured by the shift in the light they emit, known as the redshift. The velocity of a galaxy is crucial to understand because it can tell us a lot about the structure of the universe, its history, and how it might change in the future.

Studying galaxy velocities also introduces the concept of relativistic effects. As the velocities approach the speed of light, calculations based on Newtonian mechanics no longer provide accurate results. Instead, we need to apply the principles of special relativity to understand the relative motion between high-speed celestial objects.

Recessional Speed

Recessional speed refers to the speed at which astronomical objects, usually galaxies, are moving away from a given point, commonly a result of the expansion of the universe. The further away a galaxy is, the faster it appears to be receding from us; this is described by Hubble's Law.

The measurement of a galaxy's recessional speed is often done using spectroscopy, examining the redshift of the light emitted by the galaxy. As the light waves stretch out due to the galaxy's movement, the light shifts toward the red part of the spectrum, hence the term 'redshift'. By calculating the amount of redshift, astronomers can determine how fast the galaxy is moving away from us.

In the context of special relativity, when galaxies reach high recessional velocities comparable to the speed of light, we can't simply add and subtract their speeds as in classical physics. Instead, we use the formula for relativistic velocity addition to correctly calculate the speed of one galaxy as observed from another, as illustrated in the solution to the exercise provided.