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## Physics library

### Unit 16: Lesson 2

Minkowski spacetime- Starting to set up a Newtonian path–time diagram
- Visualizing multiple Newtonian path–time diagrams
- Galilean transformation and contradictions with light
- Introduction to special relativity and Minkowski spacetime diagrams
- Measuring time in meters in Minkowski spacetime
- Angle of x' axis in Minkowski spacetime

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# Starting to set up a Newtonian path–time diagram

We introduce a graph that's a lot like your good old position-vs-time graph—but with a twist!

## Want to join the conversation?

- At1:32, Sal says that "as we get more into physics, we'll see that maybe we shouldn't necessarily think of time as driving; maybe position, in some ways, is driving time." What is Sal referring to? Is there a relevant Khan Academy video or Wikipedia page?(9 votes)
- This video might help answer your question: https://www.youtube.com/watch?v=Cxqjyl74iu4

Skip to the part about light clocks.(3 votes)

- Is it that if at7:37, the photon at "c" velocity travels the same distance in less time than the spaceship, making it feel as if less time has passed? Or not? I get a little confused with relativity.(4 votes)
- For an earth-based observer, the photon in the spaceship travels a longer distance at the same speed, so that means more time must pass between ticks of the clock.

This video might help

https://www.youtube.com/watch?v=Cxqjyl74iu4(4 votes)

- hey sal, what is the meaning of word space-time?(2 votes)
- It is the three known spatial dimensions plus the dimension of time. Einstein viewed these as not separate dimensions, but all parts of the same structure known as space-time.(2 votes)

- at4:27when sal draws the graph of position of light he doesn't take into consideration the fact that the person is moving with constant velocity .(2 votes)
- Hi mohamad,

In the future videos you will learn how that the speed of light (3 x 10^8 m/s) is absolute - it is always constant no matter from what frame of reference we look at it (even if we are moving).

The Newtonium way of looking at things thought of space and time like this;

1) The passage of time is the same for all frame of references, meaning time is absolute.

2) Measured space is the same for all frames of references.

But as you will learn, this can't be.

Instead, Einstein's Theory of Special Relativity talks about how light is absolute, but space and time are not.

Now we know space and time are not absolute, they are really just different directions in this continuum spacetime. That is why people say space/the universe has 4 dimensions, the last one being time. Space = 3D and time = 1D, when you put these together you get Spacetime which is is representing that time isn't a separate absolute, and space a separate absolute, they are just different directions of spacetime.

This switch from Newtonian/Galilean thinking may take a while to sink in - just keep learning!

Hope this helps,

- Convenient Colleague(2 votes)

- I see that at huge speeds, time can depend on speed, but if you can put it either way, why switch the axes we are used to? Does this demonstration work with time as x and position as y?(2 votes)
- Yes, and yes. This is a diagram assuming Newtonian physics for the time being. It will become clearer in the following videos.(2 votes)

- what is the equation for time dilation due to speed(2 votes)
- There is no time dilation in Newtonian mechanics it is part of Special Relativity.(1 vote)

- Given that, "Inertial frames obey Newton's first law, while non-inertial frames don't. That happens because non-inertial frames are accelerating or spinning somehow, which creates pseudo forces within them, denying the first law." Why do accelerating frames create non-inertial forces actin on the accelerating masses (but they do not come from the environment they are intrinsic to the mass that is accelerating. When a body orbits the earth the earth gravity supplies the force responsible for the acceleration in the orbit,but if this is taken to be a bucket full of water the water is pushed against the bottom of the pail so it doesn't fall out becayus their is a centripetal force on the water outward and this is the so called non-inertial forse (we can feel this force when we accelerate around9are carried around relative to the ground) on a merry-go-round.(1 vote)
- There is no force pushing the water against the bottom of the bucket but there is a force from the bucket pushing on the water to make it accelerate in a circle. There only seems to be a force pushing the water against the bottom of the bucket from a non-inertial frame rotating with the bucket.

When you look at the bucket from a non-spinning frame of reference all of the forces make sense but from the spinning frame of reference there is this mysterious force pulling on the bucket and water.(3 votes)

- Why , if you change position very quickly (travel near the speed of light, for example), you move through time more slowly? I can see that the path length of the light relative to the frame the light clock is moving relative to is longer than the path length of the light relative to the frame the clock is "at rest" in, so that the time interval will be longer for the frame the clock moves relative to than the interval relative to the frame the clock is at rest in BUT why does this mean that the time flows slower relative to the frame the clock moves in , than the time flow in the clock where it is relatively at rest with respect to the frame its light path is refered to. Why couldn't we say it flowed faster than that of the clock when it was "at rest"?(1 vote)
- In special relativity the restriction that the speed of light in a vacuum is measured to be the same by all inertial observers regardless of the relative motion of the observer and the source of light requires that the time and space axes of observers in different inertial frames to have different orientations. This means that distance in space and the distance in time between two events are not necessarily the same for different inertial observers. What is the same is the total space-time distance between events. For the math to work out requires that the space-time interval to be s = (x^2)+(y^2)+(z^2)-(t^2). Because of this there is a contraction of space and time.

Here are a few good videos expanding on this. The first three are TedED videos on space-time and the 4th is from the PBS Spacetime YouTube channel:

https://www.youtube.com/watch?v=ScdLqAA_64E

https://www.youtube.com/watch?v=aeCsS6PjhK8

https://www.youtube.com/watch?v=NAXHHBUY9_E

https://www.youtube.com/watch?v=YycAzdtUIko(2 votes)

- What does sal mean by “inertial frame of reference” because it would only mean with respect to some other frame right? So I am confused at that part ,because we can’t define an absolute inertial frame of reference.(1 vote)
- An inertial frame of reference is one that follows the law of inertia: a body maintains its state of motion unless an outside force acts on it. A physicist inside an inertial frame can test for whether it is inertial by watching what happens to a pendulum, or a dropped ball, etc.(1 vote)

- What are inertial and non inertial frames of reference? Thank you.(1 vote)
- An inertial frame of reference is simply a frame of reference where the observer/object is moving at a constant velocity (i.e. undergoing no acceleration). A non-inertial frame of reference is one that is undergoing an acceleration, either speeding up, slowing down or spinning.(2 votes)

## Video transcript

- [Voiceover] Let's say
this is me right over here, and I'm drifting through space at a constant velocity relative to any other inertial frame of reference, and so I am, I am in an inertial frame
of reference myself, and in fact I'm going to define my frame of reference by me, I'm gonna say I'm at the origin of my frame of reference. So at all times, I consider myself to be stationary, and I am at the point, 'x' equals zero, and we're gonna focus on just the 'x' dimension, to simplify our discussion, and I have, you know, my oxygen and everything, and food, so no need to worry about me. Now, what I've drawn here are some axes, so that I can plot the path of things as time progresses in my frame of reference. And one thing that many of ya'll might have noticed is that I have plotted time in seconds on the vertical axis, and our position, our 'x' position, in meters on the horizontal axis. And that might be a little
bit counter-intuitive for a lot of ya'll, especially with math backgrounds, in fact, it's a little
bit uncomfortable for me, we often prefer to put time
on the horizontal axis, and position on the vertical axis, so this is a little bit counter-intuitive. But this is, you know, our choice is
a little bit arbitrary, in math class, we liked to think of our independent variable on our horizontal axis, and we think of time as
somehow driving position, so that's why we tend to put time there than we'd position there, but you can flip them around. And as we'll get more into physics, we'll see, well maybe we shouldn't necessarily think of time
always driving position, or maybe position in some
ways are driving time. But we'll get into that. But let's just first get a
little bit comfortable with this. So what would be my position over time on this diagram right over here? And you might notice these numbers, one second, two second, three second, and then in meters I have three times to the eighth, six times to the eighth, nine times to the eighth, so these are massive, massive numbers, and you could guess where
they are coming from. They are coming from the notion that the speed of light is approximately 3x10 to the eighth meters per second. But we'll get to that in a second. But let's just think about my position over time, in my frame of reference here. So, time equals zero, my 'x' position in this frame of reference is zero, I consider myself to be stationary. After one second, well my position is still zero, after two seconds my position is still 'x' equals zero, after three seconds my position is still 'x' equals zero, so my, I guess you could almost consider my path on this diagram, where I'm plotting time and space, at least in the 'x' direction, is going to look like, is going to look like, whoops, I can do a better job than that. It's going to look, it's going to look like, it's going to look like that. That would be my path on my little time and space axes, or on this little diagram that I have. But let's think about something a little bit more interesting. Let's think about me having a flashlight, so this is, I have a flashlight here, and I, at time equals zero, I turn on that flashlight, exactly at time equals zero I turn on that flashlight. And let's think about
the very first photon of light that emits from my flashlight. And let's plot it's path on my little diagram here. So time equals zero, it's exactly where I am. So time equals zero, it's position is zero meters. But after one second, where is my photon of light? And I'm gonna put it, and the photon is moving in the positive 'x' direction. So it's moving in the
positive 'x' direction with a velocity of 'C' meters per second in the positive 'x' direction. And so, after one second, it will have moved 3x10
to the eighth meters. After two seconds, it will have moved 6x10
to the eighth meters. And I've intentionally scaled my axes, so that now if I were to
draw the position of light relative to time, it's going to be at a 45 degree angle. And that's just an artifact of how I've picked my axes. But let's see if I can plot this. So it's gonna look something like this. And this is hand drawn, so it's not going to be perfect. So the position of that photon over time is going to look like that. After one second, 3x10 to the eighth meters, after two seconds, 6x10
to the eighth meters, after three seconds, we have 9x10 to the eighth meters. Alright, well that's
reasonably interesting so far, we're getting a little bit comfortable with my little diagram here, maybe some of us are getting a little bit more comfortable with time on the vertical axis, and my position on the horizontal axis. But now let's introduce another character. Another character who could almost define, or who could define their
own frame of reference. So let's say that right
at time equals zero, a spaceship passes me by. So that's, I have to draw a bigger
spaceship than that. So, there you go, that's the spaceship, and in the spaceship I have my friend, so that's the spaceship. And the spaceship is travelling with a pretty incredible velocity, in the positive 'x' direction. So I guess if this is, this vector is the speed of light, which this seems like
an awfully short arrow, but if I'm gonna say
that's the speed of light, then let's say that this thing is moving with half the speed of light. Which is still incredibly fast. So the velocity, the magnitude of the velocity here is 0.5C. So let's plot, and my friend is stationary
in that spaceship, the spaceship is moving at this constant velocity, 0.5C in the positive 'x' direction. So, they also are in an
inertial frame of reference, because it's at a constant velocity relative to me, which is an inertial frame of reference. And so let's plot my, let's plot her position on my little diagram. So time equals zero, she's exactly where I am. She's at the origin of
my frame of reference. Her 'x' position is zero. Now after one second, she's moving half the speed of light, so after one second she would have gone 1.5x10 to
the eighth meters per second. So she would have gone about that far. After two seconds, she would have gone as far as light can go in one second, because she's going
half the speed of light, so she would go about that far. After three seconds, she will have gone 4.5x10 to the eighth meters. So let me put that right over there, and you see where this is going. So if I were to plot her path, if I were to plot her path, it would look like this. It would look like this, path of my friend. That's the path of my
friend right over there. And, I want you to just
kind of think about this, think about this a little bit, think about how these different lines are, what they represent, once again get familiar with the 't' on the vertical axis. But now in the next video, we're gonna use this, what I just drew, to draw her frame of reference, to draw a coordinate system, or I guess you could say a space time diagram for
her frame of reference, on top of this one. And at least in the context of this video and the next few videos, we're going to assume a Newtonian world, we're not gonna get to Einstein in special relativity, we're just going to assume, well, we'll get to that in the next video.