Problem 29 - Fundamentals of Physics

Question

Galaxy A is reported to be receding from us with a speed of \(0.35 c .\) Galaxy B, located in precisely the opposite direction, is also found to be receding from us at this same speed. What multiple of \(c\) gives the recessional speed an observer on Galaxy A would find for (a) our galaxy and (b) Galaxy B?

Step-by-Step Solution

Answer

(a) 0; (b) approximately 0.6233c.

Step 1: Understand Relativistic Velocity Addition Formula

To find the recessional speed from the perspective of Galaxy A, we use the relativistic velocity addition formula: \ \( u' = \frac{u + v}{1 + \frac{uv}{c^2}} \) \ where \( u \) and \( v \) are the recessional velocities of our galaxy and Galaxy B from an external observer's frame, and \( c \) is the speed of light.

Step 2: Apply the Formula for Part (a) - Our Galaxy

Start by considering our galaxy's velocity as \( u = -0.35c \) from the perspective of Galaxy A. \ Using the formula: \ \( u' = \frac{0.35c - 0.35c}{1 + \frac{(0.35c)(-0.35c)}{c^2}} = \frac{0}{1 - 0.1225} = 0 \) \ Hence, Galaxy A would find our galaxy to be stationary.

Step 3: Apply the Formula for Part (b) - Galaxy B

Now consider Galaxy B's velocity, which is also \( u = 0.35c \) from the perspective of the external observer. From Galaxy A's frame: \ \( u' = \frac{0.35c + 0.35c}{1 + \frac{(0.35c)(0.35c)}{c^2}} = \frac{0.7c}{1 + 0.1225} = \frac{0.7c}{1.1225} \approx 0.6233c \) \ Therefore, Galaxy A would find Galaxy B's recessional speed to be approximately \( 0.6233c \).

Key Concepts

Special Relativity

Special relativity is a fundamental theory in physics introduced by Albert Einstein in 1905. It revolutionized our understanding of space, time, and motion. One of its key postulates is the constancy of the speed of light in a vacuum, regardless of the observer's motion. This principle challenges the way we perceive velocities and is vital for analyzing high-speed scenarios seen in cosmic scales or particle physics.

According to special relativity, the laws of physics are the same for any non-accelerating observer. Due to this, two events that occur simultaneously in one reference frame may not be simultaneous in another moving frame. This leads to time dilation and length contraction effects, which are non-intuitive but essential for understanding how motion is perceived at speeds approaching that of light. Such relativistic effects become significant when dealing with astronomical objects or particles moving at high speeds.

Velocity Transformation

In the context of special relativity, velocity transformation helps us calculate how the velocity of one object appears to another object in motion. Instead of classic addition of velocities, relativistic velocity addition must be used.

The formula for relativistic velocity addition is:

• \( u' = \frac{u + v}{1 + \frac{uv}{c^2}} \)

Here, \( u \) and \( v \) are velocities as observed from different reference frames, and \( c \) is the speed of light. This equation corrects the exaggeration of speeds that would otherwise sum up to exceed the speed of light.

For the galaxies in the problem, it allows us to determine how Galaxy A sees our galaxy and Galaxy B, despite them all moving at relativistic velocities. This is critical for understanding how observers moving at significant speeds perceive each other's motion in the universe.

Recessional Velocity

Recessional velocity refers to the speed at which a distant astronomical object is moving away from us, typically due to the expansion of the universe. It is a key concept in cosmology and helps us understand the large-scale structure of the universe.

In our exercise, both Galaxy A and Galaxy B have been observed to recede from us at 0.35c, a velocity indicating their separation due to cosmic expansion. This velocity is considered from the perspective of a stationary Earth observer, and calculating how it changes from Galaxy A's perspective requires understanding the relativistic effects.

In cosmological observations, the concept of recessional velocity is closely linked to redshift. As galaxies move away, the light they emit is stretched to longer wavelengths, revealing their speed and distance.

Speed of Light

The speed of light, represented as \( c \), is a fundamental constant of nature valued at approximately 299,792,458 meters per second. It holds a unique place in physics as the maximum speed at which information and energy can travel. This constant speed underpins Einstein's theory of special relativity.

One of the remarkable implications of the speed of light is that it remains unchanged regardless of the observer's state of motion. This leads to unusual, non-linear additions of velocities when objects move at significant fractions of \( c \), making concepts like relativistic velocity addition necessary.

In the galaxy problem, the speed of light is the benchmark that ensures the velocities remain within the bounds of physical reality. It ensures that no calculated speed exceeds \( c \), preserving causality and the structure of spacetime.

Physics Problem Solving

Solving physics problems requires a methodical approach to apply physical laws and mathematical tools correctly. Let's outline the steps using our exercise as a guideline:

1. **Understand the Problem**: Clearly define what you know (e.g., given velocities) and what you need to find (e.g., perceived velocities from another frame).
2. **Choose the Right Equations**: Select the appropriate equations or formulas. For relativistic problems, the velocity transformation formula is essential.
3. **Substitute Values Carefully**: Plug the known values into your formula, ensuring that units are consistent (e.g., using fractions of \( c \) in this exercise).
4. **Calculate**: Execute the algebra to compute the desired results. Be meticulous with steps to avoid simple errors.
5. **Interpret the Results**: Understand what your calculated values mean in the context of the problem. For instance, understanding that from Galaxy A's viewpoint, our galaxy appears stationary.

By following these structured steps, solving complex physics problems becomes more manageable and logical.