special theory of relativity tells us that one person's past may be another's future when time is relative paradoxes threaten today we peered deeper into Einstein's theory to find that the immutable ordering of cause and effect emerges when we discover the causal geography of space-time recently we've been talking about the weirdness of space-time in the vicinity of a black hole's event horizon very soon we'll be dropping below that horizon to appear at the interior of the black hole there space and time switch roles but to truly understand that bizarre statement we need to think a little bit more about how the flow of time is described in relativity today we're going to look at the amazing geometric structure that time or more accurately causality imprints on the fabric of space-time first let's recap a little bit of Einstein's special theory of relativity there are two previous episodes in particular that will be useful here if you find you need more background special relativity tells us that our experience of both distance and time are well relative if I accelerate my rocket ship to half the speed of light the distance I need to travel to a neighboring star shrinks dramatically from my point of view and observer I leave behind with an amazing Telescope observes me traveling the entire original distance they will perceive my clock is having flowed the combination of this length contraction and time dilation allows both moving and stationary observers to agree on how much older everyone looks at the end of the journey everyone agrees on the number of ticks that occurred on everyone else's clock they just don't agree on the duration of all of those ticks reminder time measured by a moving observer on their own clock is called proper time but counting those clock ticks isn't the best way for everyone to agree on space-time relationships there's this thing called the space-time interval that relates observer dependant perspectives on the length and duration of any journey that all observers will agree on even if they don't agree on the Delta X and delta T of that journey we've talked about it before but it's a tricky concept to understand intuitively but we want that intuition because more than proper time the space-time interval defines the flow of causality in relativity 3d space and 1v time become a single 4d entity called space-time to preserve our sanity we represent this on a spacetime diagram cutting time and only one dimension of space will see our causal geometry emerge plain as day even in this simplified picture there is no standing still on a spacetime diagram if I don't move through space I still travel forward in time at a speed of exactly one second per second according to my proper time clock motion at a constant velocity appears as a sloped line and the time axis is scaled so that the speed of life is a 45-degree line now let's say we have a group of space-time travelers they start at the origin where X and T equals zero they race the way to the left on the right for five seconds according to their own watches they all travel at different speeds some close to the speed of light but never faster the passed a cut through space-time is called their world line my world line is only through time and the tick marks on the time axis correspond to my own proper time clock ticks the faster a traveler moves the longer their world line that's not just because of their speed though to me their clock ticks slow they time their journey on these slow clock so I perceive them traveling for longer accounting for this we find that us-based time travelers are arranged on a curve that looks like this this shape is a hyperbola drawing a connecting line at the tickers every travelers proper time clock gives us set of nested hyperbole but these aren't just free passes these curves are kind of the contours defining the gradient of causality down which time flows and etched into space-time by the equations of special relativity to understand why we need to see how these proper time contours appear to other space-time travelers instead of doing that with equations we can feel with geometry first we need to draw the space-time diagram from the perspective of one of the other travelers to transform the diagram we need to figure out what they see as this space and time access time is easy they see themselves as stationary so their time axis is just their own constant velocity world line and their x axis well for my stationary point of view I define my x axis as a long string of space-time events at different distances but that all occurs two more taneous Li at time T equals zero to observe those points I just wait around until their light is had time to reach me at every future tick of my clock a signal arrives from the left and the right and I use that to build up a set of simultaneous events defining my key equals zero x axis our traveller does the same thing but from my point of view their clock is slow so I see them register signals at a different rate at the same time they're moving away from the signals coming from the left and towards the ones originated on the right affecting which signals are seen at a given instant the traveler infers a set of simultaneous events that to me are not simultaneous but there is no preferred reference frame their sloped x-axis is right for them even just doing this graphically we see that the travelers x-axis is rotated by the same angle as their time axis that comes from insisting that we all see the same speed of light 45 degrees on the space-time diagram moving between these reference frames is now a simple matter of squaring up our travelers axes in fact we grid up the diagram with a set of lines parallel to these new axes and square up everything while maintaining our intersection points my world line is now speeding off to the left while our traveller is motionless we just performed a Lorentz transformation but using geometry rather than mass this transformation allows you to calculate how properties like distance time velocity even mass and energy shift between reference frames the checkout what happens if I attach pens to all of the intersections when I transform between frames they trace out our hyperbola those intersections represent locations of space-time events relative to the origin they will always land on the same paper below no matter the observers reference frame I told you that these contours show where clocks moving from the origin reached the same proper time count but more generally each represents a single value for the space-time interval the Delta X and delta T of the event at the endpoint of a traveler's world line might change depending on who is watching but the hyperbolic contour that they landed on the space-time interval will not this is because the space-time interval itself comes directly from the Lorentz transformation as the only measurement of space-time separation that is unchanging or invariant under that transformation now we can finally get to why this thing is so important and what it really represents it may seem counterintuitive that an event very close to the origin in both space and time can be separated from that origin by the same space-time interval as an event that is very distant in space and time the hyperbolic shape seems to demand that but remember it takes the same amount of proper time to travel from the origin to a nearby near future event compared to a distant far future event on the same contour from the point of view of a particle communicating some causal influence those points are equivalent the space-time interval tracks this causal proximity we can think of these lines as contours on a sort of causal geography the way I define the space-time interval it becomes increasingly negative in the forward time direction so we can represent this as a valley dropping away from you here at the origin I naturally slide through time by the steepest path straight down I can change that cart by expending energy to change my velocity although doing so realigns the contours so I always slide down steepest paths there's no point anywhere down to nor that I can't reach as long as I can get close enough to the speed of light in fact the nearest downhill contour defines the forward right cone for anyone anywhere on the space-time diagram but our hill is impossible as long as the cosmic speed limit is maintained breaking that speed limit and sliding uphill are equivalent to reverse the direction of your changing space-time interval is to reverse the direction of causality to travel backwards in time the space-time diagram we looked at today was for a flat or Minkowski space in which faster than light travel is the only way to flip your space-time interval but in the crazy curved space within a black hole it gets flipped for you will soon see how this requirement of a forward causal evolution leads to some incredible predictions when we try to calculate the sub event horizon interval of space-time