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I've recently started studying the concept of space-time diagrams in special relativity, and I came across the concept of representing the time axis using $ct$, with units being that of length. Now I'm told that this is done, first of all, to keep the speed of light = 1, and give a common unit to both the axis, so that we can show, that space and time are inherently the same thing.

However, I'm still not being able to understand, how can I intuitively think of time in meters or any unit of length. Also, doesn't plugging $ct$ and $ct'$ into Lorentz transformations, make it dimensionally inconsistent. For example : $x' = \gamma(x-vt)$ if $x$ and $x'$ are in terms of distance, and so is $t$, then the term inside the bracket becomes dimensionally inconsistent.

So, how can I intuitively measure time in meters, or solve Lorentz transformations this way? What exactly does setting the speed of light $c = 1$ mean?

It would be really helpful, if someone can explain to me the motivation, and the intuition behind expressing both axis in the same units. Moreover, what would have happened, if we had kept them in separate units?

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  • $\begingroup$ Using $ct$ in the space time make the lorentz equation to be $x'=\gamma(x-\beta ct)$ where $\beta=\frac{v}{c}$. $\endgroup$
    – Iti
    Commented May 24, 2021 at 16:48
  • $\begingroup$ c has the unit $\left[ \dfrac{m}{s}\right] $ thus if unit c=1 the time has the unit meter $\endgroup$
    – Eli
    Commented May 24, 2021 at 16:51
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    $\begingroup$ Measuring time in meters instead of seconds is similar to measuring your weight in kilograms instead of newtons. $\endgroup$
    – G. Smith
    Commented May 24, 2021 at 17:51
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    $\begingroup$ If “kilograms of force” seems intuitive, “meters of time” should be intuitive. All you’re doing is using something proportional. $\endgroup$
    – G. Smith
    Commented May 24, 2021 at 17:59
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    $\begingroup$ Another way to think about this is to consider setting $c=1$ to be just a formalism. In a long calculation it can get very tedious to keep writing all the $c$’s. Physicists prefer to leave them out and (perhaps) restore them at the end, because where they need to be is clear based on dimensional analysis. $\endgroup$
    – G. Smith
    Commented May 24, 2021 at 18:15

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This question mixes a few different things. One is the idea of "natural units" where $c=1$ and another is the idea of scaling the time coordinate $t$ by the speed of light. These are completely independent.

First, notice that in units where $c=1$ you have $ct=t$. There's no magic there. Just substitution. So the question of natural units makes the scaling $ct$ trivial. The opposite is not true though. Making a graph with axis $ct$ instead of $t$ does not imply $c=1$.

Second, there's the question of making the spacetime diagrams that you raise, which could be extended to other conveniences in how to write quantities in special relativity. In any common convention for units, the quantity $ct$ has units equal to the units for measuring spatial distances. (This is "length" in a general sense, but not necessarily in units you associate with length. For example, in black hole spacetimes in general relativity, it's common to measure temporal and spatial distances in "units" of the black hole's mass, which requires units with $G=c=1$.)

If you're working in Cartesian coordinates on the spatial dimensions, then your other coordinates $x$, $y$, and $z$ also have units of length. So for some circumstances, using $ct$ instead of $t$ is handy because it emphasizes that space and time are different pats of a coherent spacetime. If you're really looking at relativistic effects, it also provides a nice scaling since, e.g., light cones run at 45 degree angles in the diagram while they would be nearly vertical or horizontal, depending on which axis you draw time on, if you used $x$ and $t$ in units like meters and seconds. But everything in these two paragraphs applies even in units where $c \neq 1$.

Third there's the question of how to make sense of the dimensionless nature of velocity in natural units. This feels strange when you're new to relativity, but it's actually something that in other contexts is probably quite familiar. The radian has the same property. It is recognized as a "unit" but it is also "dimensionless", e.g. https://physics.nist.gov/cuu/Units/SIdiagram.html. In natural units, speed has this same sort of structure.

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  • $\begingroup$ So, when we say something like 'n' units on the 'ct' axis, it essentially refers to the amount to time taken by light to travel 'n' units of distance ? In that sense, we can use distance to measure time ? Am I right ? $\endgroup$ Commented May 24, 2021 at 18:23
  • $\begingroup$ You'd be talking in that case about a distance $n$ satisfying $n=ct$, so you can solve that for $t=n/c$, which is the amount time that you described. It's probably deeper than that though. You could pick any speed $v$ and make one axis $vt$ and say you're using distance to measure time. But unless $v=c$, the speed you choose won't make this an especially meaningful axis in general. $\endgroup$
    – Brick
    Commented May 24, 2021 at 18:26
  • $\begingroup$ This choice is meaningful, because the speed of light is constant, right ? For example, someone above, gave me an analogy, that using meters to measure time, is similar to using kgf to measure weight, as kgf is the amount of force exerted by 1 kg of mass. So we are using the units of mass to measure force. Our measurement of time, is similar, right ? $\endgroup$ Commented May 24, 2021 at 18:33
  • $\begingroup$ Right. The analogy to using kg to measure weight is more like my comment about using arbitrary $v$. To use kg to measure weight, you're implicitly assuming something about the local gravitational field. In that case, that you're doing the measurement near the surface of the Earth. Going back to speed, you might find some specific problem where "non-dimensionalizing" by a velocity $v \neq c$ does make sense, but that would typically be tied to the circumstances of that problem and it having some characteristic (but maybe not fundamental) speed $v$. $\endgroup$
    – Brick
    Commented May 24, 2021 at 18:37
  • $\begingroup$ Thank you so much. Although I feel it is going to take me a while getting used to this concept. $\endgroup$ Commented May 24, 2021 at 18:43
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how can I intuitively think of time in meters or any unit of length

A meter of time is the amount of time that it takes light to travel 1 meter. A foot of time is the amount of time that it takes light to travel 1 foot. A light year of time is the amount of time that it takes light to travel 1 light year.

doesn't plugging ct and ct' into lorentz transformations, make it dimensionally inconsistent

No, although you do have to be careful. You cannot just randomly multiply by factors of $c$, you have to multiply by factors of $c/c$. So if you multiply $t$ by $c$ then you have to divide something else by $c$, typically that is velocity so that your velocity becomes a fraction of $c$.

Alternatively, if you are measuring time in units of distance then $c=1$ and furthermore $c$ is dimensionless. So in that case you can throw in factors of $c$ as desired since it is just a dimensionless constant 1. Note, this also means that all velocities are dimensionless and are given as dimensionless fractions of $c$.

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    $\begingroup$ Thank you so much. This velocity being dimensionless, and distance and time having the same units, is brought from the concept of natural units right? $\endgroup$ Commented May 24, 2021 at 20:53
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    $\begingroup$ Yes, that is correct. $\endgroup$
    – Dale
    Commented May 24, 2021 at 21:18
  • $\begingroup$ I read a little about it, for example, how cosmologists set units such that everything comes in units of energy and all, and etc. I wanted to know, is there are special motivation for doing this. Moreover, do we have to trust the math, or can we intuitively find some way or the other to explain these, like expressing length in units of energy for example ? Or should we just accept it, using the math ? $\endgroup$ Commented May 24, 2021 at 21:21

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