Consequences of Relativity: Videos & Practice Problems

Time dilation and length contraction are key consequences of special relativity. Time dilation occurs when time is measured differently in moving frames, leading to the concept that the dilated time (Δt') is always greater than the proper time (Δt₀), expressed as γ1 = 11-u2c2. Length contraction states that lengths measured in a moving frame are shorter than in the proper frame, with the contracted length (L') being less than the proper length (L₀) as L₀γ1.

Time Dilation

Time dilation is a fundamental concept in special relativity, first introduced by Einstein in 1905. It describes how time is perceived differently for observers in different frames of reference, particularly when one frame is moving relative to another. To illustrate this, consider a thought experiment involving a train. In this scenario, we have two frames: the rest frame (S) and the moving frame (S'). The moving frame is the train, which is traveling at a certain speed, while the lab frame is stationary relative to the Earth.

Imagine an observer inside the train (S') who watches a beam of light travel from the floor to a mirror on the ceiling and back down. The distance from the floor to the mirror is denoted as h. The time measured by this observer for the light to make the round trip is calculated as:

$$ t' = \frac{2h}{c} $$

where c is the speed of light. However, an observer in the lab frame (S) sees the light traveling at an angle due to the train's motion, resulting in a longer path. The light travels a distance that can be represented as the hypotenuse of a triangle, which is greater than 2h. The time measured by the lab observer is:

$$ t = \frac{2l}{c} $$

where l is the longer distance traveled by the light. Since the speed of light remains constant in both frames, the time measured must differ, leading to the conclusion that:

$$ t = \gamma t' $$

Here, γ (gamma) is known as the Lorentz factor, defined as:

$$ \gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} $$

In this equation, u represents the speed of the moving frame relative to the lab frame. The time measured in the lab frame (dilated time) is always greater than the proper time measured in the moving frame, which is the time experienced by the observer inside the train.

As the speed u increases, the value of γ increases, indicating that the dilated time becomes significantly larger than the proper time. This relationship highlights the counterintuitive nature of time in special relativity, where moving clocks tick more slowly compared to stationary clocks.

For example, consider a spaceship traveling at 11 kilometers per second, which is just below the escape velocity of Earth (11.2 kilometers per second). To determine if astronauts on the spaceship would notice time dilation, we can calculate γ using the formula:

$$ \gamma = \frac{1}{\sqrt{1 - \frac{(11 \times 10^3)^2}{(3 \times 10^8)^2}}} $$

Upon calculation, it becomes evident that the value of γ approaches 1, indicating that the time experienced by the astronauts is nearly identical to that measured by observers on Earth. Therefore, at this speed, astronauts would not notice any significant time dilation.

In summary, time dilation illustrates the profound effects of relative motion on the perception of time, emphasizing that the speed of light remains constant across all frames of reference, leading to fascinating and sometimes perplexing consequences in the realm of physics.

Time Dilation for a Muon from the Atmosphere

Muons are subatomic particles similar to electrons but with greater mass. They are produced when high-energy particles from the sun collide with atmospheric atoms, resulting in the emission of muons, which travel at approximately 90% of the speed of light. The proper time for a muon to decay, as measured in its own rest frame, is about 2.2 microseconds. This is referred to as the proper time because it represents the time interval for the decay event as experienced by the muon itself.

In contrast, an observer on Earth, who is at rest relative to the surface, will measure a longer decay time due to the effects of time dilation, a phenomenon predicted by Einstein's theory of relativity. The relationship between the proper time (\( \Delta t_0 \)) and the dilated time (\( \Delta t' \)) is given by the equation:

\(\Delta t' = \gamma \Delta t_0\)

where \( \gamma \) (the Lorentz factor) is defined as:

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

In this context, \( u \) represents the speed of the muon (0.9c), and \( c \) is the speed of light. By substituting \( u = 0.9c \) into the Lorentz factor equation, we find:

\(\gamma = \frac{1}{\sqrt{1 - (0.9)^2}} \approx 2.29\)

Multiplying this factor by the proper time gives:

\(\Delta t' \approx 2.29 \times 2.2 \, \mu s \approx 5 \, \mu s\)

This means that while the muon decays in 2.2 microseconds in its own frame, an observer on Earth measures the decay to take about 5 microseconds. This example illustrates the distinction between proper time and dilated time, emphasizing how time is perceived differently depending on the relative motion of the observer and the event being measured.

The international space station travels in orbit at a speed of 7.67 km/s. If an astronaut and his brother start a stop watch at the same time, on Earth, and then the astronaut spends 6 months on the space station, what is the difference in time on their stopwatches when the astronaut returns to Earth? Note that 6 months is about 1.577 x 10⁷ s, and c = 3 x 10 ⁸ m/s.

• A. 1.00000000033 s
• B. 0.9999999993 s
• C. 0.9999999997 s
• D. 0.0052 s

Answer: (D)

Length Contraction

Length contraction is a fundamental concept in special relativity that arises from the second postulate, which states that the laws of physics are the same in all inertial frames. This phenomenon is closely related to time dilation, where time is measured differently in various reference frames. Specifically, if time is dilated in a non-proper frame, lengths measured in that frame will be contracted compared to those measured in the proper frame.

To understand length contraction, consider measuring a rod in two different frames: the proper frame, where the rod is at rest, and a lab frame, where the rod is moving. In the proper frame, the length of the rod is measured directly, while in the lab frame, a stationary clock measures the time it takes for the rod to pass by. Due to the relative motion, the time measured in the lab frame will differ from that in the proper frame, leading to a shorter measured length in the lab frame.

The relationship between proper length (\(L_0\)) and contracted length (\(L'\)) can be expressed using the Lorentz factor (\(\gamma\)), defined as:

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

Here, \(u\) is the relative speed of the moving object, and \(c\) is the speed of light. The contracted length can then be calculated using the formula:

\( L' = \frac{L_0}{\gamma} \)

Since \(\gamma\) is always greater than 1, the contracted length (\(L'\)) will always be less than the proper length (\(L_0\)). This is the opposite of time dilation, where the dilated time is longer than the proper time.

For example, if a spaceship is measured to be 100 meters long while at rest on Earth, this length is considered the proper length. If the spaceship moves past an observer on Earth at 10% the speed of light, the contracted length can be calculated. First, we find \(\gamma\):

\( \gamma = \frac{1}{\sqrt{1 - (0.1c)^2/c^2}} = 1.005 \)

Using this value, the contracted length is:

\( L' = \frac{100 \text{ m}}{1.005} \approx 99.5 \text{ m} \)

This indicates that the spaceship appears about half a meter shorter to the observer, demonstrating that even at a significant fraction of the speed of light, the effect of length contraction is relatively small.

In summary, length contraction illustrates how measurements of length differ between moving and stationary observers, reinforcing the interconnectedness of space and time in the framework of special relativity.

Length Contraction for a Muon from the Atmosphere

In the study of relativistic physics, particularly when examining high-energy particles like muons, two key concepts emerge: length contraction and time dilation. These phenomena are interconnected aspects of the theory of relativity, illustrating how measurements of time and space can differ depending on the observer's frame of reference.

Muons, which are heavy particles similar to electrons, have a proper lifetime of 2.2 microseconds when at rest. However, when traveling at speeds close to the speed of light, specifically at 90% of the speed of light (denoted as \(0.9c\)), their observed lifetime increases due to time dilation. This effect allows them to travel further than they would if they were at rest.

To calculate the distance a muon travels in the lab frame, we can utilize the concept of length contraction. The formula for length contraction is given by:

\(L = \frac{L_0}{\gamma}\)

where \(L_0\) is the proper length (the length measured in the rest frame of the object), \(L\) is the contracted length (the length measured in the moving frame), and \(\gamma\) (the Lorentz factor) is defined as:

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

For a muon traveling at \(0.9c\), \(\gamma\) is approximately 2.29. Using this factor, we can determine the distance the muon travels before decaying. The distance can be calculated as:

\(L = 0.9c \times t\)

Substituting the time of 2.2 microseconds (or \(2.2 \times 10^{-6}\) seconds), we find that the muon travels about 594 meters in the lab frame, which is the contracted length. Conversely, if we consider the time dilation perspective, we can calculate the dilated time in the lab frame as:

\(\Delta t = \gamma \Delta t_0\)

Here, \(\Delta t_0\) is the proper time (2.2 microseconds), leading to a dilated time of approximately 5 microseconds in the lab frame. The distance traveled can then be calculated using the same velocity-time relationship, yielding a distance of about 1350 meters.

While the two methods yield slightly different results due to rounding errors, they fundamentally illustrate the same principle: length contraction and time dilation are two sides of the same coin. In essence, the proper length is measured in the frame at rest relative to the object, while the proper time is measured in the moving frame. Understanding these concepts is crucial for grasping the implications of relativistic effects on high-speed particles.

In the following figure, a right triangle is shown in its rest frame, S'. In the lab frame, S, the triangle moves with a speed v. How fast must the triangle move in the lab frame so that it becomes an isosceles triangle?

• A. 0.16c
• B. 0.84c
• C. 1.84c
• D. 1.19c

Answer: (B)

Proper Frames and Measurements

Understanding the distinction between proper frames and non-proper frames is crucial in the study of special relativity. A proper frame is defined as the frame of reference in which an object is at rest, while a non-proper frame is one in which the object is moving. This concept is particularly important when measuring lengths and times associated with moving objects.

Proper length refers to the length of an object measured when it is at rest relative to the observer. For example, if a spaceship is moving past an observer on Earth, the observer measures the length of the spaceship as it travels, which is known as the contracted length. In contrast, the proper length of the spaceship would be measured in a frame that moves with the spaceship, where it is at rest. This distinction is vital when solving problems involving length contraction, where the proper length (denoted as \(L_0\)) is always greater than the contracted length (denoted as \(L\)). The relationship can be expressed as:

\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]

where \(v\) is the velocity of the moving object and \(c\) is the speed of light.

Time dilation, on the other hand, involves comparing the time measured by two different observers. The proper time is the time interval measured in the frame where the clock is at rest, while the dilated time is measured in a frame where the clock is moving. For instance, if a muon decays in its rest frame in 2.2 microseconds, an observer in a lab frame would measure a longer time due to the effects of time dilation. The relationship for time dilation can be expressed as:

\[ \Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]

where \(\Delta t_0\) is the proper time and \(\Delta t\) is the dilated time.

In practical scenarios, such as an astronaut traveling at high speeds, the astronaut's proper time would be the time they measure during their journey, while their twin on Earth would measure a different, dilated time. This leads to the famous twin paradox, where the traveling twin ages less than the twin who remains stationary on Earth.

To effectively approach problems involving proper and contracted lengths or proper and dilated times, it is essential to identify the event of interest and determine which frame of reference is at rest concerning that event. This understanding allows for accurate calculations and predictions in the realm of special relativity.

Here’s what students ask on this topic:

What is time dilation in special relativity?

Time dilation is a phenomenon predicted by Einstein's theory of special relativity, where time is observed to run slower for an object in motion relative to a stationary observer. This effect becomes significant at speeds close to the speed of light. The relationship between the proper time (Δt₀) and the dilated time (Δt') is given by the equation:

${\mathrm{\Delta t\text{'}}}_{}=\gamma {\mathrm{\Delta t}}_0$

where $\gamma =\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}$ is the Lorentz factor, u is the relative velocity, and c is the speed of light.

What is length contraction in special relativity?

Length contraction is another consequence of special relativity, where the length of an object moving at a significant fraction of the speed of light appears shorter to a stationary observer. The proper length (L₀) is the length measured in the object's rest frame, while the contracted length (L') is measured in the moving frame. The relationship is given by:

${\mathrm{L\text{'}}}_{}=\frac{L_0}{\gamma }$

where $\gamma =\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}$ is the Lorentz factor, u is the relative velocity, and c is the speed of light.

How does the speed of light affect time dilation and length contraction?

The speed of light (c) is a fundamental constant in special relativity and plays a crucial role in both time dilation and length contraction. For time dilation, the time measured in a moving frame (Δt') is related to the proper time (Δt₀) by the Lorentz factor (γ), which depends on the speed of light:

${\mathrm{\Delta t\text{'}}}_{}=\gamma {\mathrm{\Delta t}}_0$

Similarly, for length contraction, the contracted length (L') is related to the proper length (L₀) by:

${\mathrm{L\text{'}}}_{}=\frac{L_0}{\gamma }$

In both cases, the Lorentz factor (γ) is given by:

$\gamma =\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}$

where u is the relative velocity. As u approaches c, γ increases, leading to more significant time dilation and length contraction.

What is the Lorentz factor and how is it used in special relativity?

The Lorentz factor (γ) is a key component in the equations of special relativity, describing how time and space are affected by relative motion. It is defined as:

$\gamma =\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}$

where u is the relative velocity and c is the speed of light. The Lorentz factor is used to calculate time dilation and length contraction:

1. Time Dilation: ${\mathrm{\Delta t\text{'}}}_{}=\gamma {\mathrm{\Delta t}}_0$

2. Length Contraction: ${\mathrm{L\text{'}}}_{}=\frac{L_0}{\gamma }$

As the relative velocity u increases, γ increases, leading to more pronounced effects of time dilation and length contraction.

How do proper frames and non-proper frames differ in special relativity?

In special relativity, a proper frame is a reference frame in which an object or event is at rest. Measurements made in this frame are considered 'proper' measurements. For example, the proper length (L₀) of an object is measured in the frame where the object is at rest. Conversely, a non-proper frame is a reference frame in which the object or event is in motion. Measurements made in this frame, such as the contracted length (L'), are affected by the relative motion. Proper time (Δt₀) is the time interval measured in the frame where the event occurs, while dilated time (Δt') is measured in a frame where the event is moving. Understanding the distinction between these frames is crucial for correctly applying the concepts of time dilation and length contraction.