Introduction to special relativity and Minkowski spacetime diagrams

in the last videos we constructed a little bit of a conundrum for us we had the situation where i'm drifting through space and right at time equals zero one of my friend is one of my friends she passes me by in a spaceship going half the speed of light in the positive x direction relative to me and at time equal zero is exactly where she is at my position and then she keeps traveling and so i draw the path that she is taking so after one second she would be here after two seconds she would be there and we set up our scales so that the so that on our time axis one second is the same length as on our space axis or on our path axis three times ten to the 8th meters and we did that so that at least in my frame of reference the speed of light would be at a 45 degree angle or would have a slope of 1. but we the conundrum we hit was is that the speed of light would be perceived based on this model i set up would be perceived differently based on which frame of reference you're in the photon that i emit from my flashlight right at time equals zero well sure i'm going to i'm going to see that as moving at the speed of light in the positive x direction but my friend since she's already moving in that positive x direction at half the speed of light if we assumed a newtonian world she would see that photon going at half the speed of light and likewise if she emitted a a photon from her flashlight to her it would look like it's going at the speed of light but to me it would look like it's going faster than the speed of light and the reason why that is a conundrum is we know from observation of the universe around us that this this is profound this is a little mind blowing this is counterintuitive to our everyday experience but we know from observation that the speed of light is absolute that it doesn't matter your frame your inertial frame of reference as long as your inertial frame of reference it doesn't matter your relative velocity relative to another inertial frame of reference you will always measure the speed of light at 3 times 10 to the 8th meters per second but that's at a direct contradiction with what our model just set up so it really forces us to question all of our assumptions so what are the assumptions that we made in this newtonian world well we assumed that time is absolute time is absolute and what do i mean by that well we assumed that one second passing for me is going to be one second passing for my friend that's just our everyday notion if i'm in a car and and you're not in a car and we both have watches and we synchronize our watches they seem to stay synchronized but maybe maybe we need to maybe we need to loosen this assumption we also assumed that space is absolute space is absolute and what do i mean by that well in our everyday experience regardless of what our relative frames of reference are we seem to agree that well if that that meter stick over there that's riding on that train that is a meter stick whether i'm sitting on the train or whether i'm not but maybe things start to break down a little bit as we as we think about higher velocities and maybe they break down even at the slower velocities but we just don't notice them because it's a very small error and maybe there's something even more interesting in our everyday experience we've just we just assume that time is somehow something very different than space that that you could move in the time direction without moving in the space direction or you can move in the space direction without moving in the time direction and that's and that they're independent regardless of which inertial frame of reference you're in but maybe they aren't separate things in fact maybe they're all one continuum of space time and i don't mean space dash time i mean space time i mean the word space time where space and time really aren't different things they're all there it's just one continuum space time and i keep saying it fast like that because it's not space space dash time like thinking about the different dimensions it or or thinking about two different things you're thinking we're renaming one thing called space time so these are the things that we're going to start to hold into question so what if we loosen these and we assume the thing that we have observed in the universe that the speed of light is absolute regardless of your inertial frame of reference so let's work from there and do a little thought experiment and then think about what type of a model we can construct in which case the speed of light is absolute and to do that we're still going to use our little model here and we're going to focus on my friend's frame of reference my friend who's on a spaceship who right at time equals zero is passing me by at half the speed of light and we assumed that she's in this train of spaceships so we assumed already that she's there are these spaceships that are all moving at half the speed of light in the positive x direction relative to me and they're all three times 10 to the eighth meters apart so they're like that so let's say and i did this i did these axes in blue because we're assuming this is my friend's frame of reference this you could say this is the s prime frame of reference this is sally's frame of reference from a couple of videos ago let's say a second before she gets to me and i'm also assuming the way i've drawn these axes and we're going to modify them in the future where we're actually going to use the same units for space and time right now i'm still sticking to what we've classically done where we use seconds for time and meters for our path or our space but in the future we'll actually use meters for both but we'll get to that i don't want to do too many things at once but the way i've drawn it in the hor in my along my space axis three times ten to the eighth meters is going to is going to be the exact same length as one second and i'm doing that so that so that the path of light on my diagram can be a 45 degree angle so this is 3 times 10 to the 8th meters right over there so let's say a second before sally gets to me she releases a photon from her spaceship in the direction of the spaceship that is in front of her how far is it in front of her it's 3 times 10 to the 8th meters in front of her and let's say on the back of that spaceship there is a mirror so she's she's pointing her flashlight at that mirror so what's going to happen from her frame of reference after one second so after one second from her frame of reference she is stationary so this is sally still here she's going to be still at x prime equals zero in her frame of reference and that ship is still going to be 3 times 10 to the 8th meters in front of her they're all moving relative to me at the same velocity but relative to each other they seem to be stationary and so what would be the path of that photon well that photon will have gone from sally's flashlight from her headlamp or whatever it is to that spaceship in front of her would have just reached that mirror on the back of that spaceship and so we can draw the path of light it will be so let me so that path of light will look like that on this diagram and then right at this moment right at t prime equals zero seconds the the photon will be reflected back to sally well how long will it take to get back to sally well sally is going to receive the reflection of that photon after one second because that's how long it's going to take it to go 3 times 10 to the 8th meters so the path of that very first photon the path of that very first photon is going to look like that all right well hopefully this is pretty straightforward here this is what would happen from sally's frame of reference the second before she reaches me at t equals negative one or t prime equals negative one seconds emits a photon at t equals zero seconds it gets to the spaceship that's that's three times ten to the eighth meters in front of her essentially one light second in front of her and then a second later it's what's reflecting back a second later she gets the reflection and so that's what this diagram is describing but now let's draw it projected on top of my frame of reference and this is when things are going to get really really really really interesting so i've drawn my frame of reference here and i've intentionally not marked off the seconds or the meters on my frame of reference because once again i'm not going to assume that a second in my frame of reference is a second in hers or a meter in my frame of reference is a meter in hers i have drawn her t prime axis at the same angle as i did before because for every for every light for every second we move we move into the future she's going to move half a light second in in distance in the positive x direction so this slope right here one way to think about it the way i've drawn it the slope this is a slope of two for every for every for every uh unit she moves in the x direction she will move two in the time direction and what we're going to do again is assume that on her axis i haven't drawn the x prime axis in fact this is an exercise to think about where should the x prime axis be should it be coincident with the x axis like we assume before or is it going to be in a different place but we're going to assume that the lengths i draw for one second on let's say in the s prime time in the s prime frame of reference is going to be the same as the length i would draw for 3 times 10 to the 8th meters and we're also going to assume that the speed of light is absolute so it's always going to be moving at a 45 degree angle with respect to with respect to either frame of reference so that's where things are going to get a little bit wacky but let's see let's see what's going to happen so at negative 1 seconds we still have sally turning on her flashlight she wants to bounce it off of the spaceship in front of her and so that photon is going to move with the speed of light in either frame of reference and so let me draw that let me draw that so it's going to look it's going to look like this i'm drawing at a 45 degree angle so actually i don't know where it gets reflected it's going to get reflected where it hits that x prime axis but i don't know where that is but we do know that it then gets reflected and then it gets back to sally at one second in the future so the return path of that photon is going to look something is going to look something like this and that point that it changes direction i could have let me i could have done it like this whoops i could have done it like this but the interesting point is where it changes direction because that's where that spaceship in front of her must be at that point in space-time because now we're going to start thinking of mixing up space and time but i'm not going to get too much involved in that now why is this interesting because from sally's point of view from sally's point of view this point this point here where the light changes direction from sally's frame of reference is happening simultaneously with when she reaches me this is happening at t at t prime is equal to 0 for sally so anything that is t prime is equal to 0 for sally must be on the x prime axis so this must sit on the x prime axis once again why do i know that because everything on the x prime axis any event on the x prime axis let me do a different color any i keep doing black any event on the x prime axis is going to be at t prime is equal to zero seconds and it's go from sally's frame of reference is going to be simultaneous with when she passes me up so based on that we know that this is going to be we know that we know that we know that this point which is where sally is that's going to be the origin from her frame of reference and we know that this point sits on the x prime axis so based on that we can draw the x prime axis you just need two points to define a line and so let me try to do it so let me try to draw i can do a better job than that let me try to draw the x prime axis it's going to look like it's going to look like this that right over there is the x prime axis now at this point you should find this mildly mind-blowing actually even more than mildly mind blowing because it's saying some pretty pretty crazy things first of all let's just let's just make sure we know how to read this so for any event and now we're going to start thinking in terms of space time although i'm still using different units for space and time but we'll address that in the future if i want to read its coordinates in my frame of reference well if i want to read its x coordinate i go parallel to the t axis and if i want to read its t coordinate i go parallel to the x axis but for sally's frame of reference well i essentially do the same thing if i want it's x prime coordinates i go parallel to the t prime axis and if i want the t prime coordinates i go parallel to the x prime axis but what's really interesting and i'll i'll go even deeper into this into the next video is that moment that moment right over here that from sally's frame of reference it looks like it's simultaneous with her passing me up it looks like it's happening at t prime equals zero in fact it is happening at t prime equals zero from her frame of reference it's happening at t prime equals zero from our frame of reference it is happening after sally passes up notice it is happening at t equals some positive value it's not happening at t equals zero so this is starting to get a little bit wacky one second and simultaneous time and and and and space and simul and things being simultaneous aren't going to be we're not going to agree on those depending on which frame of reference we're in the thing we will agree on is the speed of light