Minkowski Space Metric

stevendaryl
Staff Emeritus
I'm afraid your answer doesn't make much sense. I can claim that ds^2 = dx^2 - dy^2 describes a shorter path than ds^2 = dx^2 + dy^2, but I have no justification for arbitrarily flipping the sign on one of my dimensions. So why is Minkowski able to get away with it with time? Is not time orthogonal in every way to space?

The length of a path is relative to a metric. For example, if $x$ and $y$ are Cartesian coordinates, then traveling $\delta x$ in the x-direction and $\delta y$ in the y-direction will put you at a distance away from your start of $\delta s$ where $\delta s^2 = \delta x^2 + \delta y^2$. On the other hand, in polar coordinates $r, \theta$, the distance $\delta s$ is given by $\delta s^2 = \delta r^2 + r^2 \delta \theta^2$. The general notion of a metric (for 2D space, for simplicity) is a tensor, that can be represented as four numbers: $g_{11}, g_{12}, g_{21}, g_{22}$, and the corresponding notion of "distance" is given by:

$\delta s^2 = \sum_{i j} g_{i j} \delta x^i \delta x^j$

For any such metric $g_{ij}$, there is a corresponding notion of "distance" (or distance-squared, actually), and that gives rise to its own notion of "minimal" (or "extremal"; it's not necessarily minimal) path, that generalizes "shortest path" in Euclidean space. This generalized notion of a metric views distance and the corresponding notion of extremal path as something empirical, rather than something you can discover by pure logic alone. So empirically, it turns out that the metric $ds^2 = (c dt)^2 - dx^2 - dy^2 - dz^2$ is important in nature. If you move a clock from point $(x,y,z)$ at time $t$ to point $(x+dx, y+dy, z+dz)$ at time $t+dt$, then the time on the clock will advance by an amount $ds/c= \sqrt{dt^2 - (dx/c)^2 - (dy/c)^2 - (dz/c)^2}$. That's an empirical fact. The Minkowsky metric is an especially convenient way to express this fact.

The essential basis of relativity is not the universal speed limit, it is the invariance of the speed of light; the universal speed limit (and much else) follows from light-speed invariance.

I was talking to the question of "why this way" and not "how does it work". Relativity is based on the preservation of laws for all observers, unless I am mistaken - the speed limit and its universality (invariance) falls out of that.

And the problem with having no speed limit would be that there wouldn't be a spacetime metric. (I thank DaleSpam for making this more precise.)

You've just moved the "Why?" question around. Why do we live in a universe that has a universal speed limit instead of one that does not?

My observation was that the question is more constrained, not simply formulated as loose but differently. Causality is essential.

There's a fine and consistent mathematical model for describing a universe in which the speed of light is not invariant and there is no universal speed limit; it's called classical mechanics and there's nothing wrong with it except that observation tells us that it's not the way the universe works.

Interesting. I haven't thought of it that way.

The obvious problem is that it is a nonphysical approximation as you note, which we now know can't be realized. E.g. fields are quantum relativistic, not classical infinite-speed. Most problematic would be the loss of a working cosmology, I think.

I note that there would also be loss of generality, without the universal speed limit c the electric and magnetic field would "fall apart", et cetera.

And of course space, time and causality comes unglued as per my first point and are tacked on as ad hoc constraints rather than a (more or less approximate) map of a physical system (spacetime in GR).

But this has become philosophy and not science, so I will stop there.

I'd like to thank stevendary,lpervec andtNugatory for their informative posts. What I get from this is Lorentz looked at his data and asked 'how would time and space need to be shaped in order to explain these observations?' and from there we have the minus sign on the metrics for space. I can live with that. However, I don't see how we escape the conclusion that space is imaginary.

Fredrik
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I'd like to thank stevendary,lpervec andtNugatory for their informative posts. What I get from this is Lorentz looked at his data and asked 'how would time and space need to be shaped in order to explain these observations?' and from there we have the minus sign on the metrics for space. I can live with that.
I wouldn't say that it has anything to do with "shape", because Minkowski spacetime is completely flat, in the sense of differential geometry. (The Riemann tensor is zero everywhere). I prefer to think of it this way: The result that pervect explained suggests that the function g defined below is going to be useful.

However, I don't see how we escape the conclusion that space is imaginary.
$$\eta=\begin{pmatrix}-1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix},$$ and then define ##g(x,y)## for all 4×1 matrices ##x,y## by ##g(x,y)=x^T\eta y##. We haven't used any complex numbers in the definition of the function ##g##, and the right-hand side of that last equality is still equal to ##-x^0y^0+x^1y^1+x^2y^2+x^3y^3##.

I'm using units such that c=1. If you don't, you might want to take the upper left component of ##\eta## to be ##-c^2## instead of -1. Alternatively, you can take the "zeroth" component of x (i.e. ##x^0##) to be c times the time coordinate, rather than the time coordinate.

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robphy
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I'd like to thank stevendary,lpervec andtNugatory for their informative posts. What I get from this is Lorentz looked at his data and asked 'how would time and space need to be shaped in order to explain these observations?' and from there we have the minus sign on the metrics for space. I can live with that. However, I don't see how we escape the conclusion that space is imaginary.

Then I shouldn't suggest a related experiment:
Galileo does the experiment I proposed (with a limited range of velocities (say, up to the speed of the fastest horse)),
then extrapolates the portion of his "circle" to infinite velocities. What would be the equation of Galileo's circle?

You might not recognize that this diagram is essentially the position-vs-time graph drawn and interpreted in every introductory physics class... It's just that its non-euclidean geometry is not treated [or recognized]).

(In these two experiments without fancy equations, I have actually produced the metric of Minkowski spacetime and the degenerate-metric of Galilean spacetime. If I continue my story, I can build up all of the geometry of Special Relativity and Galilean Relativity.)

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pervect
Staff Emeritus
Then I shouldn't suggest a related experiment:
Galileo does the experiment I proposed (with a limited range of velocities (say, up to the speed of the fastest horse)),
then extrapolates the portion of his "circle" to infinite velocities. What would be the equation of Galileo's circle?

You might not recognize that this diagram is essentially the position-vs-time graph drawn and interpreted in every introductory physics class... It's just that its non-euclidean geometry is not treated [or recognized]).

(In these two experiments without fancy equations, I have actually produced the metric of Minkowski spacetime and the degenerate-metric of Galilean spacetime. If I continue my story, I can build up all of the geometry of Special Relativity and Galilean Relativity.)

I think I'm missing something. I get ##R = v t = v \tau## for the Galliean case, and ##R = v / \sqrt{1-(v/c)^2} \, \tau## for the relativistic case, but I don't see how to use this to derive relativity, rather I use relativity to derive the results. Here ##\tau## is the proper time for which the observers run, in your example it's a constant (one minute), but I've given it a symbolic value anyway.

Nugatory
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I3What I get from this is Lorentz looked at his data and asked 'how would time and space need to be shaped in order to explain these observations?' and from there we have the minus sign on the metrics for space.

There's more to the history than that.

Well before Einstein and as early as 1895, Lorentz developed the coordinate transformations that were consistent with the null result of the Michelson-Morley experiment. None of this stuff about metrics, geometry, space-time intervals showed up in this formulation; it was just an alternative to the Galilean transforms, one in which the ##\gamma## constant showed up and time did something a bit more complicated than the Galilean ##t'(x,y,z,t)=t##.

In 1905 Einstein demonstrated that these Lorentz transformations can be derived from the principle of relativity and the light-speed invariance. That introduced no new mathematics, but established those two principles as the basis for all subsequent theoretical physics.

Two years later, in 1907, Minkowski recognized that the Lorentz transformations were mathematically equivalent to a geometry in which the metric took on the form diag(-1,1,1,1) or diag(1,-1,-1,-1) depending on one's choice of sign conventions. That's when the metrics/geometry/space-time interval stuff appeared. At first it seemed to be just a more abstract mathematical formulation of what Einstein had already discovered, but it turned out to be essential to making the next jump to general relativity.

I can live with that. However, I don't see how we escape the conclusion that space is imaginary.
Easy... use the other sign convention, which is really nothing more than a trivial coordinate transformation, and space won't be "imaginary". Of course then time will be, but the ease with which I can flip them with a simple mathematical trick suggests that there is no physical significance to the complex numbers that appear when I take the square root of squared intervals calculated using the Minkowski metric.

Here's a more prosaic example of a mathematical formalism leading to a conclusion that you ought to be able to escape no matter what the math says: Standing at a height ##H## above the ground, I throw a ball upwards with speed ##v##. How many seconds later does the ball strike the ground? This is a fairly standard high-school sort of problem... but when we solve it, we find (because we're dealing with a quadratic equation) that we have two solutions, one positive and one negative. We could look at the negative time solution and say that we're stuck with the inescapable fact that the ball can travel backwards in time and strike the ground before we threw it. Or we can say that just because we can calculate a negative time doesn't mean that we have to assign any physical significance to it.

robphy
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I think I'm missing something. I get ##R = v t = v \tau## for the Galliean case, and ##R = v / \sqrt{1-(v/c)^2} \, \tau## for the relativistic case, but I don't see how to use this to derive relativity, rather I use relativity to derive the results. Here ##\tau## is the proper time for which the observers run, in your example it's a constant (one minute), but I've given it a symbolic value anyway.

The idea is this: suppose we really did these experiments in the real world [without a theory yet to explain the observation]. How could one obtain a theory to explain it? Let's pretend that Euclid, Galileo, and Minkowski performed these experiments.

If Euclid set up surveyors on a plane and told them, from a point, travel in various directions and stop when their odometers read 1 mi, what is the locus of these stopping points? A circle, of course. Suppose each surveyor also had a long ruler, which they somehow carried "perpendicular to his radial path". Each could assign coordinates to points: t [along his path] and x [perpendicular to his path.... defined by being tangent to the circle]. Each surveyor would make a map of the stopping points such that t^2+x^2=(1 mi)^2, all identical--independent of surveyor. From this we get a way to measure the separation between points.

Now to find a separation between surveyor paths (radial lines), Euclid defines an angle as the arc-length intercepted divided by the radius of the circle. Define cos(angle) as the ratio between the t-coordinate of the stopping point of the other radial line and the radius of the circle (that is, the t-coordinate of the stopping point on my path). sin(angle) is the ratio between the y-coordinate and the radius of the circle. tan(angle)=slope=y/t. Note that since arc-length is additive, then angle is additive but slope as tan(angle) is not. You are now on your way to deriving the rest of Euclidean geometry.

If Galileo did this on position-vs-time graph with a wristwatch and a ruler, he would get his version of a "circle" by extrapolating an apparently vertical segment (by probing a small range of slow speeds) to a vertical line t^2+(0)x^2=(1 minute)^2. Note that all tangents to the circle agree... so they agree on elapsed times between events (i.e. absolute simultaneity). Galileo's version of cosine would equal 1 (no time dilation) and Galileo's version of slope (a.k.a. velocity) would coincide with Galileo's version of "angle" (Galilean rapidity)... so velocities and angles are additive. With some work (and a spatial metric), you could get the Galilean spacetime geometry.

Of course, Minkowski's version of a "circle" would be a hyperbola t^2+(-1)x^2=(1 minute)^2, with asymptotes x=t and x=-t. Galileo couldn't see the hyperbola from his extrapolation from the small speed range. However, Minkowski had access to really fast particles. Alas, the tangents no longer agree as in Galileo's case (simultaneity is not absolute) and velocities are no longer additive. Minkowski's version of cosine is greater or equal to 1 (i.e. time dilation) and would be identified as the hyperbolic-cosine (a.k.a. gamma). With some work, you could get the Minkowski spacetime geometry.

There's more to the history than that. [..]

Two years later, in 1907, Minkowski recognized that the Lorentz transformations were mathematically equivalent to a geometry in which the metric took on the form diag(-1,1,1,1) or diag(1,-1,-1,-1) depending on one's choice of sign conventions. That's when the metrics/geometry/space-time interval stuff appeared. At first it seemed to be just a more abstract mathematical formulation of what Einstein had already discovered, but it turned out to be essential to making the next jump to general relativity.

[..]
Just a little nitpicking (see my post #11): the geometric invariant space-time interval stuff appeared already in 1906 with a paper by Poincare paper that described the Lorentz transformation as a 4-dimensional space rotation (in the section on gravitation).

I'd like to thank stevendary,lpervec andtNugatory for their informative posts. What I get from this is Lorentz looked at his data and asked 'how would time and space need to be shaped in order to explain these observations?' and from there we have the minus sign on the metrics for space. I can live with that. However, I don't see how we escape the conclusion that space is imaginary.

I have to agree with Copernicus, I don't see what we gain by pretending that time (or space) isn't imaginary. Depending on how you set up your delta S squared, it could be either one. You can try to explain that away somehow, but I think Minkowski's trick of replacing the Y axis with ict and treating the Lorentz transformation as a rotation is the most instructive approach to understanding the invariant interval.

robphy
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I have to agree with Copernicus, I don't see what we gain by pretending that time (or space) isn't imaginary. Depending on how you set up your delta S squared, it could be either one. You can try to explain that away somehow, but I think Minkowski's trick of replacing the Y axis with ict and treating the Lorentz transformation as a rotation is the most instructive approach to understanding the invariant interval.

Minkowski's trick is useful, if you stop with Special Relativity.
If you wish to move to on General Relativity (or do Special Relativity with different coordinates), you'll likely get stuck.
In the end, using the metric $g_{ij}$ is the best way to go.

There is a famous passage in Misner Thorne Wheeler's Gravitation:
"Farewll to ict"

• dextercioby
Btw, I remember reading that section and not being convinced, although now I'm going to have to re-read it. It's frustrating to think that I'm going to have to master GR before I can make a qualified comment on this subject, seeing as it seems obvious that you have to treat either time or space as imaginary. Which one is it?

Fredrik
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it seems obvious that you have to treat either time or space as imaginary. Which one is it?
You definitely don't have to treat any of them as imaginary, since (as I have explained twice) the minus sign(s) can be explained in other ways. And the choice of where to put the minus signs has no relevance whatsoever. You can view the two choices as defining two different theories of physics if you want to, but those theories make exactly the same predictions, so they should at least be viewed as equivalent.

robphy
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This is interesting to note... but it's likely not relevant to the current discussion:

"Where the sign of the metric makes a difference"
Phys. Rev. Lett. 60, 1599 – Published 18 April 1988; Erratum Phys. Rev. Lett. 60, 2704 (1988)
Steven Carlip and Cécile DeWitt-Morette
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.60.1599

You definitely don't have to treat any of them as imaginary, since (as I have explained twice) the minus sign(s) can be explained in other ways. And the choice of where to put the minus signs has no relevance whatsoever. You can view the two choices as defining two different theories of physics if you want to, but those theories make exactly the same predictions, so they should at least be viewed as equivalent.

Your explanation is unsatisfactory. You seem unaware that moving a negative sign into a metric doesn't change the fact that either time or space when squared is negative in the formula for distance in order to create an invariant distance of 0. The only interpretation for this is that either space or time is imaginary.

Fredrik
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Your explanation is unsatisfactory. You seem unaware that moving a negative sign into a metric doesn't change the fact that either time or space when squared is negative in the formula for distance in order to create an invariant distance of 0. The only interpretation for this is that either space or time is imaginary.
Where are you guys getting that idea? It's false. You only have to use imaginary numbers if you insist on using the Euclidean inner product (i.e. the dot product) instead of some other bilinear form. The significance of that minus sign (or equivalently, those three minus signs) is just that a different bilinear form is more useful than the dot product in this theory.

dextercioby
Homework Helper
I don't know if this is worth mentioning, but people outside HEP use the - + + + signature instead of the + - - - one, because not only it allows you to go to an arbitrary number of spatial dimensions without changing the negative value of the determinant, but also because in 4 dimensions the Hamiltonian formulation would mean to disinguish between up spatial indices and down spatial indices. If the signature is + + + on the spatial part, then you can use only <downstair> indices.*

And to make a personal note which seems on topic: I really wish editors would republish some books (valuable ones) whose authors used the ict. The most imporant to me would be: Quantum Field Theory by Umezawa and P.Roman's Theory of Elementary Particles.

*Note: Just after saying that, I noticed that the famous ADM 1962 seem to use + - - - :)

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robphy
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Your explanation is unsatisfactory. You seem unaware that moving a negative sign into a metric doesn't change the fact that either time or space when squared is negative in the formula for distance in order to create an invariant distance of 0. The only interpretation for this is that either space or time is imaginary.

As hinted in my post #30
with that reasoning, you might conclude something similar [or worse?]
for good-ol' Galilean physics's position-vs-time graph from PHYSICS 101,
which would have $ds^2=dt^2+0^2(dx^2+dy^2+dz^2)$ or ($ds^2=0dt^2+(dx^2+dy^2+dz^2)$, if you prefer the spatial metric).

As Fredrik said, you are led into such mis-interpretations
if you insist on thinking that you are dealing with a Euclidean Geometry [which you are not].

Dale
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The only interpretation for this is that either space or time is imaginary.
This is simply factually wrong. You have been given a different interpretation in posts 2, 3, 4, 29, 36, and 39.

A.T.
I've never seen a satisfactory explanation of the metrics used in a calculation of distance in Minkowski space. In Euclidean space, the distance is:
ds^2 = dx^2 + dy^2 + dz^2
But in Minkowski space, the distance is
ds^2 = (dt * c)^2 - dx^2 - dy^2 - dz^2
Why are the signs reversed? This implies that space (or time depending on your convention) is imaginary.

If you don't like the negative signs in the metric, you should look into space-propertime diagrams, which are Euclidean:

robphy
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If you don't like the negative signs in the metric, you should look into space-propertime diagrams, which are Euclidean:

I think it's been mentioned before that such diagrams are not position-vs-time diagrams, as drawn in PHY 101 and in Special Relativity courses.

It's not a faithful mapping of events where each event is represented by one point on the map (as seen on your twins animation... the reunion event appears as two points on the "propertime"-axis).

In addition, light-rays that cross at an event in a Minkowski diagram (i.e. a portion of a light-cone)
are shown as (non-intersecting) parallel segments in the space-propertime diagram.
So, I think a light-clock will look very strange on this diagram.

A.T.
I think it's been mentioned before that such diagrams are not position-vs-time diagrams, as drawn in PHY 101 and in Special Relativity courses.
Yes, to pros and cons of both type of diagrams are discussed in the thread I linked.

This is simply factually wrong. You have been given a different interpretation in posts 2, 3, 4, 29, 36, and 39.

I've yet to read an interpretation in any of these posts that changes the geometry of an invariant distance that's at the foundation of Minkowski space. That is,

0 = (dt * c)^2 + dx^2 + dy^2 + dz^2

The metrics are simply semantic ways to reorganize this relation. At the heart of all of these matrices and formulas is the basic, physical reality that this distance plus this distance plus this distance plus this distance is equal to zero. The only way I know of to make this work is for one of the distances to be negative (or all of the distances to be zero). A negative distance has no real-world equivalent.

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Nugatory
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At the heart of all of these matrices and formulas is the basic, physical reality that this distance plus this distance plus this distance plus this distance is equal to zero.

As Fredrik pointed out above, no one is sayng that "this distance plus this distance plus this distance plus this distance" is equal to zero. We are saying that a particular bilinear form on those four quantities (which are not distances, but functions of differences of coordinates) is zero.

It may be time to close this thread, as we are are repeating ourselves without effect.