Problem 15 - Fundamentals of Physics

Question

The center of our Milky Way galaxy is about 23000 ly away. (a) To cight significant figures, at what constant speed parameter would you need to travel exactly 23000 ly (measured in the Galaxy frame) in cxactly \(30 \mathrm{y}\) (measured in your frame)? (b) Measured in your frame and in lightyears, what length of the Galaxy would pass by you during the trip?

Step-by-Step Solution

Answer

(a) Travel at 0.999956c. (b) 30 light-years pass by in your frame.

Step 1: Understanding the Problem

We need to find the speed parameter at which an object would travel exactly 23000 light-years as measured in the Galaxy frame in 30 years measured in the traveler's frame. We've also got to calculate what length of the galaxy would pass by from the traveler's frame.

Step 2: Identify Required Relativistic Equations

This problem requires the concept of time dilation from special relativity, where the time dilation factor (gamma, \(\gamma\)) is given by:\[\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\]where \(v\) is the velocity of the object and \(c\) is the speed of light.

Step 3: Calculate Time Dilation Factor (\(\gamma\))

The time measured in the proper frame (30 years in the traveler's frame) relates to Galaxy's time frame as \(\Delta t = \gamma \Delta t_0\), where \(\Delta t_0 = 30\) years. Given Galaxy frame time \(\Delta t = 23000\) years,\[\gamma = \frac{23000}{30}\]

Step 4: Calculate the Speed Parameter \(v\)

Using the time dilation factor calculated, and substituting it in the formula for \(\gamma\), we solve for \(v\):\[\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} = \frac{23000}{30}\]Solving for \(v\), we square both sides,\[1-\frac{v^2}{c^2} = \left(\frac{30}{23000}\right)^2\]and solve for \(v\):\[v = c \sqrt{1 - \left(\frac{30}{23000}\right)^2}\]

Step 5: Calculate the Length of the Galaxy in Traveler's Frame

The contracted length \(L\) as measured in the traveler's frame is given by:\[L = \frac{L_0}{\gamma}\]where \(L_0 = 23000\) ly (length in Galaxy's frame).Using the previously calculated \(\gamma = \frac{23000}{30}\), we have\[L = \frac{23000}{\frac{23000}{30}} = 30\text{ ly}\]

Step 6: Verify the Solution

Ensure the calculated values of \(v\) and contracted length \(L\) are consistent with initial conditions and logically valid according to the theory of relativity.

Key Concepts

Time Dilation

Time dilation is a fascinating concept from Einstein's theory of special relativity. It describes how time can appear to pass at different rates depending on your relative motion.
When you travel at a significant fraction of the speed of light, time for you effectively "slows down" compared to someone who is stationary. This means that if you're moving fast enough, you could travel long distances while aging very little. This is exactly what's happening when we talk about traveling 23000 light-years in just 30 years, as measured in the traveler's frame.
The time dilation formula is:

• \(\Delta t = \gamma \Delta t_0\)

where \(\Delta t\) is the dilated time, \(\Delta t_0\) is the proper time, and \(\gamma\) is the gamma factor. Get the gamma factor right, and you unveil the magic of time travel—or, at least, the closest we can get according to the laws of physics!

Length Contraction

Length contraction is another intriguing idea from special relativity. It explains how objects contract along the direction of motion when observed from a stationary reference frame.
Imagine traveling in your spaceship across the galaxy. To you, the galaxy doesn't look as vast as it does to someone standing still in their frame. This is because, at such high velocities, the lengths in the direction of travel shorten. So, for you, the entire 23000 light-years stretch of the Milky Way appears just 30 light-years long!
The length contraction formula is:

• \(L = \frac{L_0}{\gamma}\)

where \(L\) is the contracted length, \(L_0\) is the original length, and \(\gamma\) is the gamma factor. Length contraction will make long journeys feel much shorter when traveling at relativistic speeds.

Velocity of Light

The velocity of light, denoted by \(c\), is one of the few constants of nature that holds an extraordinary place in physics. Light travels at approximately 299,792,458 meters per second (or about 186,282 miles per second), and it's the ultimate speed limit of the universe. Nothing with mass can reach—or exceed—this velocity.
This constant plays a crucial role in special relativity equations, such as those for time dilation and length contraction. Thus, whenever we are calculating relativistic effects, the velocity of light is the reference point against which all other velocities are measured.
In our exercise, understanding that no object can travel faster than light is fundamental when solving for the speed parameter of our galaxy-traveling spaceship. This boundary opens up the realm of relativistic effects, demonstrating how velocities just under \(c\) can lead to significant time dilation and length contraction.

Gamma Factor

The gamma factor, \(\gamma\), is a crucial component of relativistic equations in special relativity. It's derived from the Lorentz transformation and accounts for the effects of both time dilation and length contraction.
Gamma factor is defined as:

• \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\)

where \(v\) is the velocity of the object and \(c\) is the speed of light. As your velocity approaches the speed of light, the gamma factor increases exponentially.
This means that the faster you go, the more pronounced the effects of time dilation and length contraction become.
In the exercise, the gamma factor helped us deduce how much time would pass for a traveler moving at near-light speeds and how the distance they travel would appear contracted. It's a remarkable demonstration of how relative motion can radically alter our perception of time and space!