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When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ ds^{2} = -(cdx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} $$ where $ x^0,x^1,x^2,x^3 $ come from $ x^{\mu} : \mu = 0,1,2,3 $, and $ c $ is the speed of light. The first resource I had access to--I have to do a bit of digging for the exact paper--of course addressed that this invariant takes the tensor form: $$ ds^2 = g_{\mu \nu}dx^{\mu}dx^{\nu} $$ as well. This two things I've seen in all of my texts and online resources regarding resources. The element in question that varies between authors, is the time coordinate, $ x^0 $. When it was first explained to me, it was using the standard Cartesian representation for the spatial portion of the coordinates, and the time coordinate was labeled as $ x^0 = ict $. Squaring gives $ (x^0)^2 = -c^2t^2 $, and applying differential calculus, we get $ (dx^0)^2 = -c^2dt^2 $. Sensible and expected to have the tensor formula spit out the Minkowski Metric. The author then later explicitly states the coordinates are $ x^0 = ict, x^1=x,x^2=y,$ and $x^3 = z$.

My questions is then, why is it most authors on the subject omit the imaginary unit on the time coordinate? For example, here.

The only reason I can fathom omission is if the author is using metric signature $ [+,-,-,-] $, where I started off learning the theory with signature $ [-,+,+,+] $ which may be the reason seeing the time coord with no imaginary unit seems dissonant to me. All help appreciated!

Edit: After reading the other answers, my questions is now one of why and how (mathematically) do we obtain the Minkowski Metric Signature. More specifically the one element with a different sign.

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Sort of related: en.wikipedia.org/wiki/Wick_rotation –  BMS Apr 8 at 18:08
    
Very helpful, but still not getting to the root of why :C –  Doryan Miller Apr 8 at 18:10
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Really, the root answer is that you don't get much by adding all of the machinery of complex numbers -- what is the conjugate of a spacetime point, and what does it mean? Should inner products be $v^{*}\cdot v$? Why is the time coordinate the only one that takes a complex number? If you say "there's just a minus sign" in the signature, you avoid these problems. –  Jerry Schirmer Apr 8 at 18:47
    
Hmmm. So from what I understand, the assumptions and "guesses" that we make in mathematical physics are literally just that. The means to an end? –  Doryan Miller Apr 8 at 18:56

5 Answers 5

up vote 11 down vote accepted

as you wrote, the spacetime invariant can be expressed as: $$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$$ and from that we normally get: $$ds^2=-c^2dt^2+dx^2+dy^2+dz^2$$ This is not because of some arbitrary imaginary time unit, this is because the metric ($g_{\mu\nu}$) is a diagonal matrix with the coefficients of each term of the $ds^2$ equation: $$g_{\mu\nu}=\left(\begin{array}{l}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}\right)$$ and the coordinates are listed as you would assume: $$dx^{\mu}=\left(\begin{array}{l}cdt\\dx\\dy\\dz\end{array}\right)$$ Then, you should note that $$g_{\mu\nu}dx^{\mu}=dx_{\nu}=\left(\begin{array}{l}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}\right)\left(\begin{array}{l}cdt\\dx\\dy\\dz\end{array}\right)=\left(-cdt~~dx~~dy~~dz\right)$$ Also, $v_{\mu}v^{\mu}$ is the inner product, meaning: $$dx_{\nu}dx^{\nu}=\left(-cdt~~dx~~dy~~dz\right)\left(\begin{array}{l}cdt\\dx\\dy\\dz\end{array}\right)=-c^2dt^2+dx^2+dy^2+dz^2$$ This is the equation you want without any imaginary unit omission. The reason for the $-1$ in the $g_{\mu\nu}$ is that it makes the system Lorentz invariant; it maintains $ds^2$ as a spacetime invariant quantity.

Let me be historical. In Euclidean 3-D coordinates, you find the interval between positions as $$\Delta d_{Eucl}^2=(X_2-X_1)^2+(Y_2-Y_1)^2+(Z_2-Z_1)^2$$ When incorporating relativity and time, the interval becomes a spacetime quantity. Because relativity sets the maximum speed of information as $c$, we make the interval $$\Delta s^2=\Delta d_{Eucl}^2-c^2(t_2-t_1)^2$$ This represents the original interval - the distance between the two events - minus the maximum distance the information could travel in the time between the two events. That difference lets us determine if the events happened in a definite chronological order ($\Delta s^2<0$) or if they occurred in two distinctly separate positions ($\Delta s^2>0$), since in relativity we can't always be sure. It is from this that the $-1$ in the metric originates. Space and time coordinates are given opposite signs here. We keep the metric in terms of $s^2$ because we simply can't be sure if $s$ is positive or negative. There was no original imaginary time coordinate, that was simply someone's poor interpretation and it has been (thankfully) dropped for the most part.

I should probably also point out that the imaginary time coordinate can not come out of Euclidean 4-D either. If one ignores relativity, then there is no maximum velocity. If there is no maximum velocity, there is no natural way of equating spatial and temporal coordinates. Therefore, not only would it not be right to use $c$ in the $ict$ coordinate, it also would not make sense to add time to space because there would be no agreeable conversion between them. However, if you don't ignore relativity, then you must subtract the time term from the 3-D interval in order to comply with the notion of a maximum velocity. So the Euclidean signature, $(1,1,1,1)$ can not be used to describe 4-D spacetime! So you never define the time coordinate as imaginary.

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I was under the impression that the metric has that negative 1 because of the imaginary unit, but this is clearly not the case. Is there a more rigorous derivation of the choice of metric signature? I hate to believe it was arbitrary and just worked. –  Doryan Miller Apr 8 at 18:06
    
Sadly, it was arbitrary and just worked. Well, no, it wasn't arbitrary. When Einstein wrote his original paper on special relativity he had to write time with an opposite sign to to space, but he didn't write the metric in the way we do these days. He was more driven by the physics. When Minkowski reformulated SR we got the metric with the signature we use today. –  John Rennie Apr 8 at 18:12
    
Have links or names to any Minkowski papers off the top? ( I'm searching now ) –  Doryan Miller Apr 8 at 18:13
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It was not arbitrary at all, it is from the spacetime interval. In Euclidean space, an interval is the sum of the squares of the difference between the three coordinates of two positions.... –  Jim Apr 8 at 18:14
    
when you introduce relativity, you take that interval and find the difference between it and the distance light could travel in that amount of time, which is $s^2=d_{Eucl}^2-c^2t^2$. That effectively finds the spacetime separation between two events. This is done because relativity takes $c$ as the max speed, so the true interval when incorporating time would have to be the measured interval minus the maximum interval in the given time –  Jim Apr 8 at 18:17

If you define $x^0=ict$, then I assume one takes $x_\mu=x^\mu$ so that the metric is actually $\eta_{\mu\nu}=\text{diag}(1,1,1,1)=\delta_{\mu\nu}$, i.e. you're dealing with a Euclidean metric. Then $$ds^2=\delta_{\mu\nu}dx^\mu dx^\nu$$ gives the usual outcome : $$ds^2=-c^2dt^2+d\vec{x}^2$$ The usual conventions are as follows:

Option one: One defines $x^\mu=(ct,\vec{x})$ and $x_\mu=\eta_{\mu\nu}x^\nu=(-ct,\vec{x})$ where $\eta_{\mu\nu}=\text{diag}(-1,1,1,1)$. This results in $$ds^2=-c^2dt^2+\vec{x}^2$$This convention is usually taken in treatments that focus on (general) relativity and/or spacetime structure.

Option two: One defines $x^\mu=(ct,\vec{x})$ and $x_\mu=\eta_{\mu\nu}x^\nu=(ct,-\vec{x})$. where $\eta_{\mu\nu}=\text{diag}(1,-1,-1,-1)$. This results in $$d\tilde{s}^2=-ds^2=c^2dt^2-\vec{x}^2$$ This approach is usually taken when the focus is on particle physics results. My personal theory is that this is because it results in the equation $p_\mu p^\mu=m^2$ (as opposed to $p_\mu p^\mu=-m^2$), which is one of the main equations in particle physics and arguably looks slightly more pleasing than the alternative. As was pointed out in a comment, it also makes time-like intervals positive, which might be preferred when dealing with particles (which, of course, always travel along time-like or light-like trajectories).

Of course, the two approaches are completely equivalent to each other. The convention with the imaginary time coordinate has fallen out of use a little bit. I can see why this would happen: It's not a particularly helpful convention for intuition, nor does it seem to carry over well to general relativity.

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I love it when the answer is convention! Thank you very much for clarifying that! –  Doryan Miller Apr 8 at 18:02
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I think particle physicist choose that signature because it makes timelike intervals positive, which is nice if you are dealing with mostly particle trajectories. –  George G Apr 8 at 18:09

In general, the expression for the metric and the expression for the coordinates need to work together to give you the correct line element. So the following combinations all have the same line element $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$:

$g_{\mu\nu}=diag(1,1,1,1)$ with $x^{\mu}=(ict,x,y,z)$

or

$g_{\mu\nu}=diag(-1,1,1,1)$ with $x^{\mu}=(ct,x,y,z)$

or

$g_{\mu\nu}=diag(-c^2,1,1,1)$ with $x^{\mu}=(t,x,y,z)$

The primary reason to abandon the "ict" notation and use one of the last two is because we eventually want to go beyond special relativity and do general relativity. That is best done using (pseudo-) Riemannian geometry, and Riemannian geometry requires real-valued coordinates. Furthermore, when using Riemannian geometry you are not even guaranteed to have a time coordinate, so you cannot easily be sure where to put the "i".

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"... my questions is now one of why and how (mathematically) do we obtain the Minkowski Metric Signature. More specifically the one element with a different sign."

Well, If you check Einstein's "Relativity: The Special and General Theory", you will find in the Appendix I (just before equation (10)) that Einstein simply started off with the Pythagorean Theorem in 4D, which he put like this:

$$r = \sqrt{x^2 + y^2 + z^2} = ct$$

He then squared both sides and moved $c^2t^2$ to the left changing the sign. This gave him the signature [-, +, +, +] Obviously, you are equally free to move the $x^2 + y^2 + z^2$ to the right, in which case you will get the signature reversed [+, -, -, -] (apparently, that's what Minkowski did).

Curiously enough, from the fact that Einstein used the Pythagorean Theorem we can see that he is talking about lengths/distances here, and not coordinates (which are points) as he later keeps calling them (and everybody after him). On the other hand, it is quite obvious even without looking at the derivation, if you come to think about it. Simply, you cannot shrink points (coordinates) into infinitesimals like dx or dy, since they cannot get any smaller then they already are - their extension is exactly null. You cannot also square points to make them $x^2$, $y^2$, $z^2$. Shrunk point and squared point is still the same point. At the same time, ds is called the "line-element" (Einstein called it also "linear element"). Line suggest distance or length, and not a point (or coordinate for that matter), doesn't it?

And by the way, this equation implies that time "travels" orthogonally to x, y, z. Sure, we do draw t orthogonally to x for instance, in order to better visualize various functions, like acceleration. But when we observe physical things, like a car, accelerate without changing direction, they do not "draw" a hiperbole in space, do they?

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The only real reason to introduce $ict$ coordinates is to stress the similarity (for didactic purposes I guess) between Lorentz transformation and orthogonal rotations in more used-to Euclidian space.

Note that Minkowski pseudo-Euclidian space obtains exactly "normal" Euclidian form if complex time is introduced,namely: metric signature becomes $++++$: exactly like if it was regular Euclidian 4-space. Also, more vivid: matrix of Lorentz transformation obtains exactly form of real orthogonal matrix due to $\cos(ix) = \cosh x$ (similar for $\sin x$). Thus, you rotate through complex angle, but matrix looks like regular orthogonal, for example boost in $x$-direction, $$ \left( \begin{matrix} \cos z & \sin z & 0 & 0\\ -\sin z & \cos z & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{matrix} \right) $$

where $z$ is now strictly imaginary.

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