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Measuring time in meters in Minkowski spacetime

As counterintuitive as it might sound, it will become extremely convenient to use the same units for space and time. Happily, we can do that just by multiplying the time axis by c, the speed of light. Here's how.

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Video transcript

- [Voiceover] I am about to do something that most of you will probably find disconcerting, in fact, the first time that I saw someone do this, I too found it disconcerting. So, just as a little bit of background, so far, when we've been thinking about our space axis, and we've really just been focusing on one dimension of space, we've been focusing on the x dimension, or the x prime dimension, depending on whose frame of reference we are talking about. We measured that in terms of meters. We measured that in terms of meters, and then when we thought about time, well, we said, initially, we said, "Well, time is "fundamentally something different than space," so we had a different set of units called seconds, in fact, you know, this goes well before we, well before the 20th century when we got special relativity, this goes all the way back to well before even that, where we said time is something different. We measure it in units of time, in seconds, or minutes, or hours, or days. While distance, which is different than time, we measured in meters, or feet, or miles, or kilometers, or whatever else. But as we started to get into this world of special relativity and we started to see that space and time are not each absolute. And, in fact, we can actually think of all events as happening in this continuum called spacetime. And I say it fast, "spacetime". Because it's not saying "space-time". Two different things. It's saying that they're really just different directions in this continuum spacetime. And so, if they are all the same thing why do we use different units for space and time? Why do we use seconds for time and meters for space? Or at least in the examples that I've been doing so far. And so to fix that, instead of calling this the "t-prime axis" or the "t axis," instead of labeling it in terms of seconds, what we can do and this is the part that many of you will find disconcerting, let's call this the "c times t-prime axis." And let's call this the "c times t axis." And the "c times t-prime axis." Well, what's that going to do? Well, we know that the speed of light is an absolute. It is, if we are measuring it in, and I'll do approximately because it's actually two point nine something. But, approximately three times 10 to the 8th meters per second. For the sake of all of these videos I'm just assuming it's three times ten to the 8th meters per second. For simplicity it's roughly that. So if we were to take c times time instead of this being one second, in terms of seconds, well, we multiply it times three times ten to the 8th meters per second. Well, the seconds cancel out and we're left... If we want to measure time in terms of meters, would be three times ten to the 8th meters. So this is three times ten to the 8th meters, instead of calling it one second. This over here is negative. So that over there is negative three times ten to the 8th meters. Along the "c t-prime axis." Likewise, this what we called one second, right over here, instead we can call this as three times ten to the 8th meters. This we could call negative three times ten to the 8th meters. Now this will be counterintuitive to you because you've always viewed time as fundamentally something different than meters. You haven't been thinking in terms of spacetime. In fact, in our normal human experience we don't experience the world in terms of space time. Time is something that we are just falling forward into and space is something that we feel like we have more agency and we can move in the different dimensions of space more easily, while time just feels like we're plummeting forward in that dimension. But now we're thinking in terms of spacetime and this makes our units the same. Now if it helps you, you could view this as three times ten to the 8th light meters. So you could think of this as the time it would take for light to go three times ten to the 8th meters. Likewise, if we wanted everything in terms of what we traditionally conceive of as our time dimensions, we could have kept this as one second, instead of calling this three times ten to the 8th meters we could have called this one light second, which would be the distance that light travels in what we measure, or what we consider to be one second. But the benefit of this is now we're consistent. We're measuring different directions in spacetime, which is kind of a continuum, there is no separate time and space. We're measuring them all in the same units, which we will find is very, very convenient. So, I know this is going to take a little bit of time to get your head around, and I'll maybe do a few more videos to make you feel comfortable with this type of idea, but this will be a convenient thing for us. Especially as we start to have a metric in our, what we would call, our Minkowski spacetime. Because we are going to be operating in the different dimensions as if they have the same units. So, hopefully, I encouraged you to kind of sit and ponder and think about this a little bit, and hopefully it doesn't bother you too much.