Relativity: how people get time dilation wrong

Summary

This script delves into Einstein's theory of special relativity, tackling the common misconceptions and paradoxes that arise from time dilation. It emphasizes the importance of understanding the Lorentz factor and the relativity of simultaneity, illustrating how the perception of time can vary depending on the observer's motion and position. The video aims to clarify the subtleties of relativity, urging viewers to revisit the Lorentz transform equations for a deeper comprehension of time's relativistic nature.

Takeaways

• 🕰️ Time dilation is a fundamental concept in special relativity, where moving clocks appear to tick slower compared to stationary ones.
• 🧐 The Twin Paradox is a famous example in special relativity that raises questions about the consistency of time dilation.
• 📚 The Lorentz factor (γ) is crucial in understanding time dilation; it equals 1 for stationary objects and is greater than 1 for moving objects.
• 🔍 The Lorentz factor is calculated as \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \), where \( v \) is the velocity of the moving object and \( c \) is the speed of light.
• 🤔 The concept of time dilation can seem paradoxical because it implies that observers in relative motion measure different elapsed times for the same event.
• 🔄 Relativity requires that the laws of physics are the same for all observers, regardless of their state of motion, which can lead to apparent contradictions.
• 📐 The Lorentz transforms are essential for translating between the perspectives of two observers moving relative to each other, including both time and position.
• 📉 The time experienced by a moving observer depends not only on their own time but also on the location of the event in their frame of reference.
• 🤹‍♂️ Comparing clocks between two moving observers is complex and requires careful consideration of both time and location.
• 📚 It's important to understand the context and meaning of equations in relativity to avoid misinterpretations and false paradoxes.
• 🔑 The Lorentz transform equations are fundamental to special relativity and provide clarity when concepts seem counterintuitive or paradoxical.

Q & A

What is the main focus of the video script on Einstein's theory of special relativity?

The video script focuses on explaining the concept of time dilation in special relativity, the Lorentz factor, and the importance of being careful when discussing paradoxes in relativity, such as the Twin Paradox.

What is the Lorentz factor and how is it related to the speed of an object?

The Lorentz factor, denoted as gamma, is a factor unique to relativity that equals one for a stationary object and is greater than one when an object is moving. It is calculated as one over the square root of one minus the square of the ratio of the object's velocity (v) to the speed of light (c).

What is the time dilation equation as presented in the script?

The time dilation equation is t_sub_moving = gamma * t_sub_stationary, where t_sub_moving is the time experienced by someone who sees the clock moving, and t_sub_stationary is the time experienced by someone who doesn't see the clock moving.

Why is it important to understand the Lorentz transforms when discussing time dilation?

The Lorentz transforms are important because they show how to translate between the points of view of two observers who are moving with respect to each other. They provide a more general understanding of time dilation and help avoid misconceptions or paradoxes.

What is the Twin Paradox mentioned in the script, and why is it considered a paradox?

The Twin Paradox is a thought experiment in special relativity where one twin travels at high speed on a spaceship to a distant star and ages more slowly than the twin who remains on Earth. It is considered a paradox because it seems to violate the principle of relativity that all observers should measure time in the same way.

Why does the script suggest that comparing clocks between two moving observers is tricky?

Comparing clocks between two moving observers is tricky because the time experienced by each observer depends not only on their relative motion but also on their respective locations. This requires careful consideration of the Lorentz transforms and the specific clocks being compared.

What does the script imply about the importance of understanding the context of equations in relativity?

The script implies that it is crucial to understand the context and meaning of equations in relativity to avoid making mistakes or drawing incorrect conclusions. Simply grabbing equations from textbooks without understanding their implications can lead to misunderstandings.

How does the script address the issue of ambiguity when two observers claim to be stationary?

The script addresses this issue by explaining that both observers are equally right in claiming to be stationary, and that the laws of physics should be the same for both. It suggests that understanding the Lorentz transforms and the specific conditions under which time dilation equations apply can resolve this ambiguity.

What is the significance of the script's statement that 'physics is everything'?

The statement 'physics is everything' signifies the importance of a deep understanding of physical principles, especially in the context of relativity, where seemingly paradoxical phenomena can be explained through careful application of physical laws.

How does the script suggest one should approach learning about relativity?

The script suggests that one should approach learning about relativity with caution, always referring back to the Lorentz transform equations for clarity, and not being too quick to accept or dismiss ideas without thorough understanding.

Outlines

00:00

🕰️ Time Dilation and Relativity Paradoxes

This paragraph introduces Einstein's theory of special relativity, focusing on the concept of time dilation where clocks moving relative to an observer appear to tick more slowly. It discusses the public's fascination with relativity and the common misconceptions that arise, such as the Twin Paradox. The script emphasizes the importance of understanding the math behind relativity to avoid paradoxical claims. The Lorentz factor, denoted as 'gamma', is introduced as a key element in time dilation equations, highlighting its role in determining the time experienced by a moving observer relative to a stationary one.

05:03

🔍 Delving into the Lorentz Transforms and Time Dilation

The second paragraph delves deeper into the Lorentz transforms, which are fundamental equations in relativity for translating between the viewpoints of two observers in relative motion. It clarifies that the time dilation equation presented earlier is a special case of a more general equation, which includes an additional term related to the relative velocity and position. The importance of considering the location of the observer when analyzing time dilation is underscored. The paragraph uses the Lorentz transform to demonstrate how the time experienced by a moving observer (Observer 2) differs depending on whether the comparison is made at the location of a stationary observer (Observer 1) or at the moving observer's own location.

10:03

🤔 The Subtleties of Comparing Time in Relativity

The final paragraph emphasizes the complexity and subtlety of comparing time between two observers in relative motion. It points out that without a proper understanding of the relativity equations, one might incorrectly conclude that there is a paradox. The paragraph stresses the importance of revisiting the Lorentz transform equations for clarity and accuracy. It concludes by encouraging viewers to think critically about relativity and to share the knowledge gained from the video to help others understand the careful application of relativistic principles.

Keywords

💡Special Relativity

Special Relativity is a theory developed by Albert Einstein that fundamentally changed our understanding of space and time. It suggests that time and space are relative and not absolute. In the context of the video, it's the foundation for the discussion on time dilation and the Twin Paradox, illustrating how moving clocks can appear to tick slower than stationary ones.

💡Time Dilation

Time Dilation is a direct consequence of Special Relativity, where time appears to pass slower for an object in motion relative to a stationary observer. The video script uses the concept to explain how interstellar travelers would age more slowly than those remaining on Earth, a key element in the narrative of the video.

💡Twin Paradox

The Twin Paradox is a thought experiment in Special Relativity that highlights the counterintuitive nature of time dilation. It involves a pair of twins, where one travels at high speed and ages less than the other who remains on Earth. The video mentions this paradox to emphasize the complexities and seeming paradoxes that arise from relativistic effects.

💡Lorentz Factor

The Lorentz Factor, denoted as 'gamma' in the script, is a mathematical expression used in Special Relativity to account for the relativistic effects such as time dilation and length contraction. It is defined as one over the square root of one minus the square of the velocity divided by the speed of light squared. The video explains its importance in calculating the time experienced by a moving observer.

💡Lorentz Transforms

Lorentz Transforms are the mathematical equations that describe how to convert the space and time coordinates of an event from one inertial frame of reference to another. In the video, they are used to explain the relationship between the observations of two observers moving relative to each other, particularly in the context of time dilation.

💡Interstellar Travel

Interstellar Travel refers to the hypothetical journey between stars, which is a common theme in science fiction and a real possibility in theoretical physics. The video uses this concept to illustrate the implications of time dilation, where a traveler to a distant star would experience time differently than someone on Earth.

💡Relativity Paradoxes

Relativity Paradoxes are situations that arise from the predictions of Special Relativity that seem to contradict common sense or classical physics. The video script discusses the need to be careful when claiming paradoxes in relativity, emphasizing the importance of understanding the underlying principles.

💡Clocks

Clocks in the script are used metaphorically to represent the passage of time as experienced by different observers in relative motion. The video explains how clocks 'tick' at different rates depending on their state of motion, which is a central theme in understanding time dilation.

💡Velocity (v)

Velocity, denoted as 'v' in the script, is a measure of an object's speed in a given direction. It plays a crucial role in the Lorentz Factor and the time dilation equation, determining the extent to which time slows down for a moving observer relative to a stationary one.

💡Speed of Light (c)

The Speed of Light, denoted as 'c' in the script, is the ultimate speed limit in the universe according to Einstein's theories. It is a fundamental constant used in the equations of Special Relativity, including the Lorentz Factor, and is central to understanding the limits of motion and the behavior of time and space.

💡Ambiguity

Ambiguity in the script refers to the apparent contradictions that arise when considering the relative motion of observers and the effects on their measurements of time. The video emphasizes the need to carefully define the reference frames and the clocks being discussed to resolve these ambiguities.

Highlights

Einstein's theory of special relativity has been a subject of fascination for a century, with peculiar effects such as time dilation and length contraction.

The concept of moving clocks ticking more slowly is foundational to the idea of time experienced differently in interstellar travel.

The Twin Paradox in special relativity is a classic example that raises questions about the theory's consistency.

The time dilation equation, t_moving = γ * t_stationary, is central to understanding the relativistic effects on time.

The Lorentz factor (γ) is key in relativity, relating to the motion of objects and affecting time and space measurements.

The ambiguity in relativity arises when considering which observer is stationary, affecting the perceived rate of time.

Relativity requires that the laws apply equally to all observers, regardless of their state of motion.

The Lorentz transforms are essential for translating between the viewpoints of two observers in relative motion.

Time dilation is a special case of the more general Lorentz transform equation, highlighting its dependency on specific conditions.

The time experienced by an observer depends on both their relative motion and their spatial position in their own frame of reference.

Careful definition of clocks and their locations is crucial when comparing time between observers in different frames.

The Lorentz transform for time reveals that the time seen by one observer depends on the time and position in the other observer's frame.

Observers in relative motion will perceive time differently at different locations, emphasizing the complexity of time dilation.

The video emphasizes the importance of understanding the context and implications of relativistic equations before applying them.

The Lorentz transform equations are the foundation for resolving apparent paradoxes and misunderstandings in relativity.

The video concludes by urging viewers to approach relativity with caution and a deep understanding of its principles.

Transcripts

Einstein’s theory of special relativity has fascinated the public for about a hundred

years.

People hear about clocks running at different speeds, objects shrinking, and a myriad of

peculiar other effects.

And people might be willing to accept those sorts of assertions, but when they start thinking

about relativity, they come up with what seems to be paradoxes. And then they begin to doubt

the theory, because, well, you know- paradoxes.

One such paradox is the oft-quoted statement that moving clocks tick more slowly than stationary

ones. This is the whole basis of the ideas of interstellar travel, where a person heading

at high speed on a spaceship to a distant star ages more slowly than a person stuck

on Earth. If that rings a bell, that’s the basis of the famous Twin Paradox of special

relativity.

I’ll actually make a video focusing entirely on the Twin Paradox, but in this video I want

to really drive home some math that shows that you have to be ultra, mega, extremely

careful when you claim that there is a paradox in relativity.

So let’s start with the most common time dilation equation you’ll find in a relativity

textbook. It is t sub moving equals gamma times t sub stationary.

T sub moving is the time experienced by someone who sees the clock moving, T sub stationary

is someone who doesn’t see the clock moving and gamma is a factor unique to relativity.

It's called the Lorentz factor. It is equal to one for something not moving and greater

than one when something is moving. Gamma is simply one over the square root of the quantity

one minus v squared over c squared. That'll be important in a little bit, but, for the

moment, just remember that gamma is greater or equal to one.

At this point, it's probably helpful for you to watch two other videos of mine, specifically

the one on the derivation of the gamma factor and the second one introducing Einstein’s

equations. I won’t assume that you’ve seen them as I go through the rest of this

video, but what I say here will a bit clearer if you've got those two under your belt.

Okay, so let’s get back to the time equation. What it says is simple. Given that gamma is

greater than or equal to one, it means that the time experienced by a person seeing the

clock move is more than the time experienced by someone who doesn’t see it move. And

this means that someone moving ages more slowly than a person who isn’t. Now this sounds

crazy, but it has been proven to be true. And I made a video about that as well. It’s

called Einstein’s Clocks.

So this idea is behind a lot of science fiction and it’s true to a degree. But it’s not

the entire story. There are two key points I want to make here.

The first is that this makes absolutely no sense. Not because the two clocks tick at

different rates- although that’s pretty weird- but because of an ambiguity.

Suppose you label the two people as number 1 and number 2. If we take number 1 as seeing

the clock to be stationary and number 2 as seeing it moving, we see that number 2 experiences

more time than number 1.

However, relativity requires that the laws are the same, no matter who is not moving,

and this poses a problem. Suppose that we say number 2 is stationary with respect to

his clock and number one says that the clock is moving. Then number one experiences a longer

time.

Okay, so this is a problem. If you pick person one to be stationary with the clock, then

person two’s clock ticks faster. If you pick person two to be stationary with the

clock, then person one’s clock ticks faster. And both people are equally right in saying

that they are stationary. They can’t both have clocks ticking faster than the other.

That just makes no sense at all.

Either we have logically proven that relativity is just bogus, or there's something more to

it. Given that the scientific community still embraces relativity, there has to be something

more to it. And the answer is both very subtle and technical. So let’s dig into that.

So, begin by putting up the most basic and general equations of relativity. These are

called the Lorentz transforms and they show how you can translate between the point of

view of two observers who are moving with respect to one another. Let’s put up the

time dilation equation I mentioned before as well.

Notice that the Lorentz transforms have both a position and time equation. We’re not

interested in the position one here, so we’ll get rid of it and we’ll change the symbols

for the time one so it looks more like you usually see in the textbooks.

So the first thing you see is that the time dilation equation is a special case of the

more general one. They're the same only if that extra term of v over c squared times

x sub stationary equals zero. And that is true only if either the velocity v is zero

or x sub stationary is zero. The first one is pretty trivial, as that means that the

two people aren’t moving with respect to one another and so you don’t need relativity

theory at all. But the second is a lot trickier. It says that the time experienced by the person

seeing the clock moving depends both on the time experienced by the person not seeing

the clock move and by the location in the frame of the person who sees the clock to

be stationary.

So that’s a big point. What it means is that we have to be very careful about defining

which clock we’re talking about. There are many clocks stationary with respect to each

observer. Since they are at different locations, we need to take that into account.

Since this is a core point of this video, I want to explicitly show what this means.

Let’s start out by just talking about one observer, which we’ll call number 1, who

is stationary with respect to the clock. We draw him here with his clock.

But that’s not the only clock in his world. He has clocks at every location and, since

those clocks aren’t moving with respect to him, all of those clocks are ticking identically.

Now let’s add in observer 2. He sees observer 1 moving to the right at velocity v and he

wants to know what he thinks the moving clocks will read.

To do that, we will use the Lorentz transforms but, to do that, we need to figure out the

locations. Let’s pick two locations- say the location of observer 1 and observer 2-

as seen by observer 1. Observer 1 always thinks his location is x sub 1 equals 0.

Since observer 1 thinks observer 2 is moving to the left at velocity v, he sees observer

2’s position as changing. According to observer 1, observer 2’s position is just x sub 1

equals minus v times t sub 1.

Let’s stop the motion and concentrate just on the two locations and times as seen by

observer 1. His own location is simply x sub 1, t sub 1 equals 0, t sub 1.

Now let’s write down observer 2’s location as seen by observer 1. It is just x sub 1,

t sub 1 equals minus v times t sub 1, t sub 1.

We can now use the Lorentz transform for time. Let’s first find out what Observer 2 thinks

the time is at observer 1’s location. We just put in the values seen by observer 1.

In doing so, we find that t sub 2 equals gamma t sub 1. That’s just the normal time dilation

equation.

Now let’s do the same thing for how observer 2 sees what observer 1 sees at observer 2’s

location. Remember that observer 1 sees this location as x sub 1, t sub 1 equals minus

v t sub 1, t sub 1.

We start with the equation t sub 2 equals gamma times the quantity t sub 1 plus v over

c squared times x sub 1. We put in the x and t seen by observer 1.

Thus we get that t sub 2 equals gamma times the quantity t sub 1 minus v squared over

c squared times t sub 1. So that’s pretty easy.

We can factor out the t sub 1 and get this equation here. And at this point, we can use

the definition of gamma, which I remind you is one over the square root of the quantity

one minus v squared over c squared. That means that this equation can be written as t sub

2 equals gamma, divided by gamma squared, times t sub 1.

And finally, we get t sub 2 equals t sub 1 divided by gamma. That’s divided, not multiplied

by. That’s exactly the opposite thing we saw when we looked at what was happening at

observer 1’s location.

Now we need to step back and review a bit. This means that the transformation for what

observer 2 sees compared to what observer 1 sees depends on the location. Observer 2

sees more time elapsed than Observer 1 sees at Observer 1’s location, but sees less

time elapsed than Observer 1 sees at Observer 2’s location.

This is a super subtle point, but it clearly says that comparing clocks between two moving

observers is a tricky business. You need to take into account both the time each experiences

and the location at which the comparison is going on. Further, they both agree that the

remote person’s clock is slower. You probably need to think long and hard about this and

maybe even rerun this video to get your head around this idea.

A crucial point here is that when people start grabbing equations from relativity textbooks

without understanding exactly what the equations mean, then they’ll almost certainly make

a mistake. And perhaps the most important point is that if you ever- and I mean ever-

find something about relativity that sounds weird to you, always go back to the Lorentz

transform equations. They’ll never let you down.

Okay, so that might have been a mind-bender, but I hope it taught you not to be cavalier

about using relativity. And, of course, I welcome your comments and hope you’ll like

the video and subscribe to the channel. Plus share the video with your friends so we can

help people understand that you need to be careful with relativity. In the meantime,

remember- physics is everything.