# Imaginary part in Ds

Bonjour,

Je suis nouveau à ce forum. :-) I am a newbie in this forum..

My question concerns the imaginary part in the relativistic distance. Is someone has some physical interpretations? Why should this sqrt(-1) exist in this equation? Not the mathematical explanation :-( but the physical interpretation!

Thanks

Imagine...

Gold Member
Bonjour! J'ai essayé d'écrire ceci en FranÃ§ais
,mais mon FranÃ§ais, c'est tres mal!

An imaginery separation corresponds to a 'space-like' seperation, meaning that the events are outside of each others light cones.

quartodeciman
That is a good question. I suspect it will have a rather long and convoluted answer.

Complex numbers had to be fully established as a mathematically valid and interesting thing before such use would become widely accepted in physics. This took place mainly during the nineteenth century. So, it basically started as a new direction in mathematical analysis. Acceptance of complex numbers in physical scientific work took a lot of convincing. This was pretty much an accomplished fact by the twentieth century.

Nevertheless, Albert Einstein used no "i"s in his first definitive publications on relativity theory. He was mathematically conservative at the time, even though he had taken a course in Fonktiontheorie (which means complex-valued function calculus) in college. His faith and will to rely on more advanced mathematical tools only came after 1912, when he collaborated with his college chum Marcel Grossman. Herman Minkowski was the first (I believe) to use complex number notation in the subject of relativity around 1908. Minkowski had been one of Einstein's college teachers, but Einstein still stuck to conventional mathematical representation until he got completely bogged down in details of a general theory of relativity and needed to plea for help from his mathematician friend (MG).

What is the answer to your question? I am as anxious to learn as you. My suspicion is that it is tied to the philosophical subject of just how mathematics is supposed to be associated with physical science (facts) at all. The nineteenth century posed many problems that subsequently yielded to analysis with complex numbers and functions: harmonic analysis, optical wave mechanics and, yes, Maxwell's electromagnetic theory. One answer we will probably hear a lot is: it works, so don't complain about 'meanings'.

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Gold Member
The answer really isn't that convoluted, basically when a separation between two events in spacetime is complex this is a 'spacelike' separation (becasue it's like the separation between two events that have 'no time' separation but a spatial separation i.e. two events happening simulataneously (ignoring the failure of simultaneity at distance)). This means that the two events may not effect each other.

Tyger
Originally posted by Imagine
Bonjour,

Je suis nouveau à ce forum. :-) I am a newbie in this forum..

My question concerns the imaginary part in the relativistic distance. Is someone has some physical interpretations? Why should this sqrt(-1) exist in this equation? Not the mathematical explanation :-( but the physical interpretation!

Thanks

Imagine...

It's really nothing more than an unfortunate choice of notation that was made in the early days of Relativity, and it has caused countless confusion. None of the physical quantities are "imaginary", and few people use that notation anyway.

Staff Emeritus
Gold Member
Sorry, but the answer is mathematical. The formulae for proper time squared (d&tau;2) and proper length squared (dl2) are just the negatives of each other (in units where c = 1), so it's convenient to use just one formula for ds2 and then compute time and length in terms of ds. If you take the square root of both sides, then one of them will wind up with an i in it... but as long as you don't try to compute the length of a time interval or the duration of a length, the results will always be real.

Bonjour,

Thanks jcsd, I know concerning 4D squared distances.

Thanks quartodeciman for the historical context, really appreciated. And, I agree with you, it shall be a long and convoluted answer. That's why I ask it in this forum.

Tyger, "None of the physical quantities are imaginary" is a BIG assumption. I would appreciated to have more explanation.

Hurkyl, would it mean that physicists are able to "observe" only squared distance (4D) and we use unsquared distance in math to solve toward whatever and therefore introduce root(-1)?

Imagine

fando
I'll provide another answer, which I hope helps people understand.

The imaginary component ict in special relativity is just an artifact of the math chosen to work with the theory. In Gravitation, by Thorne, Wheeler et al, they do away with this in favor of math that doesn't use ict. There should be nothing philosophically interesting in why time is treated as an imaginary space coordinate, it's just something that comes with framing things a certain way. There is an analogy to be made with the imaginary numbers encountered in circuit diagrams and equations. Here's how Gravitation starts off describing this:

Farewell to "ict"

One sometime participant in special relativity will have to be put to the sword: "x4 = ict." This imaginary coordinate was invented to make the geometry of spacetime look formally as little different as possible from the geometry of Euclidean space; to make a Lorentz transformation look on paper like a rotation; and to spare one the distinction that one otherwise is forced to make between [certain quantities in tensor math].

In other words, in flat spacetime, using ict is sensible, but it can't really be used in a physics of curved spacetime.

Staff Emeritus
Gold Member

I'm going to repeat myself, but this time with details, so it might make more sense!

The formula for proper time is:

(c d&tau;)2 = (c dt)2 - dx2 - dy2 - dz2

The formula for proper length is:

(dl)2 = -(c dt)2 + dx2 + dy2 + dz2

For paths corresponding to timelike seperation, the formula above for proper time yields a positive value, and thus we can take the square root and get a real number for time. For paths corresponding to spacelike seperation, the formula above for proper length yields a positive value, and thus we can take the square root to get a real number for length.

Because the formulae are so similar, it is convenient to introduce a new quantity we call the "metric":

ds2 = (c dt)2 - dx2 - dy2 - dz2

And then we have the formulae:

d&tau;2 = (1 / c2) ds2
dl2 = - ds2

Or if we prefer the nonsquared versions:

d&tau; = (1 / c) ds
dl = -i ds

For timelike paths, d&tau; is a positive real number. For spacelike paths, dl is a positive real number (because ds is imaginary).

ds is a computational and theoretical convenience that allows us to manipulate both d&tau; and dl simultaneously, by writing formulae for ds and using the conversions above to get d&tau; and dl. Complex numbers are an intermediary that makes our work easier, but it all works out to a real number in the end.

Bonjour,

Fando:
1) "in flat spacetime, using ict is sensible"! Would you explain "sensible"? (excuse my english)
2) "imaginary numbers encountered in circuit diagrams"! If I remember, the presence of imaginary part in electronic circuit induce oscillating behavior. No?
3) I understand that the introduction of imaginary part is a mathematical artifice but could it be an artifact conducting toward something else, physically interpretable?

Hurkyl:
Two occurences of "convenien..."! Infinitesimal calculations are also convenient but subject to physical interpretation and full of physical signifiances.

Would it mean that physicists are able to "observe" only squared distance in 4D?

Staff Emeritus
Gold Member
Two occurences of "convenien..."! Infinitesimal calculations are also convenient but subject to physical interpretation and full of physical signifiances.

Infinitessimal calculations also describe things that ordinary calculations cannot; they make mathematical powers of expression more powerful.

The use of ds is simply to unite two similar formulae to reduce the work we need to do. ds is always an observable; over a temporal duration, ds is simply the speed of light times the elapsed proper time. Over a spatial dislpacement, ds is simply i times the proper length.

Anyways, if you're really fishing for a physical interpretation, you should look into other geometries. One interesting one in particular is the hyperbolic plane, whose oddities are intimately related to the reasons i appears in that one formula, and the geometry of Special Relativity in general. (note: I think the hyperbolic plane as I'm going to use it has some relation to hyperbolic geometry, but I don't know for sure)

The Euclidean plane is usually described as the x-y plane, but I'm sure you know of polar coordinates (r, &theta;); the conversion between them is, of course:

(x, y) = (r cos &theta;, r sin &theta;)

We imagine the r-coordinate being the size of a circle and the &theta;-coordinate being the angular position along that circle.

Well, we can do a similar kind of thing with the hyperbolic trig functions, though it's not quite so nice! We have to imagine three standard kinds of hyperbolas; ones that open left-right, ones that open up-down, and degenerate ones that are just two diagonal lines. The formula for this class of hyberbolas is:

x2 - y2 = a

If a is positive, then we have left-right opening hyperbolas, if a is negative we have up-down opening hyperbolas, and if a is zero we have a pair of diagonal lines.

Just like with circles, any point of the plane is on one of these hyperbolas, and we can write hyperbolic coordinates for a point of the plane. Because we have 3 different kinds of hyperbolas, there are three kinds of conversions:

(x, y) = (r cosh &phi;, r sinh &phi;)
(x, y) = r (+/-1, 1)
(x, y) = (r sinh &phi;, r cosh &phi;)

(here r can be both positive and negative)

This plane can be described, IIRC, in a way similar to how complex numbers work. Any number of the hyperbolic plane can be written as (x + h y) where h2 = 1 (not -1), and you can derive properties for these hyperbolic numbers that are similar to those of the complex numbers. These numbers even have some utility; I've seen them used in a short derivation of the cubic formula (though I don't remember how to do it)... but be careful because I don't think you can, in general, divide by a hyperbolic number.

The connection between this and the oddity with the metric goes back to the defining equation for the class of hyperbolas:

x2 - y2 = a

I wrote it like that for simplicity of exposition; I'm sure you're aware that when writing the equations for standard conics, the parameter a should have been squared. The equations for the hyperbolas should be:

x2 - y2 = r2
or
y2 - x2 = r2

But if you don't want to have this proliferation of equations, you can instead write it as:

x2 - y2 = r2 where r is either purely real or purely imaginary.

r here is analogous to the metric ds just like the r in polar coordinates is analogous to the ordinary Euclidean metric.

The quaternions may be something interesting to study as well.

Guybrush Threepwood
Originally posted by Imagine

2) "imaginary numbers encountered in circuit diagrams"! If I remember, the presence of imaginary part in electronic circuit induce oscillating behavior. No?

"induce" is not the most correct word. If you have sinusoidal currents/voltages it's easier to use imaginary numbers. You can resolve the circuit without them but that would be REALLY hard. Also impedance has an imaginary part for the same reason. You could have resistance, inductance and capacitance but again it's more easy to use an impedance that contains them all.

Bonjour,

GuyBrush:
"induce" is not the right word but with a smily ...

Hurkyl:
Nice explanation about hyperbolic coordinates but that's math stuffs. I am interested in physical stuffs. For example, IMHO, derivatives and integrals are physiccally full of meaning before being math stuffs.

Still physically interrogating!

Staff Emeritus
Gold Member
Well, I was trying to skip down to how the hyperbolic plane relates to the issue at hand.

And, unfortunately, I don't really know of a good introduction to it on its own merit; you can find an introduction to it by studying special relativity and space-time diagrams, but those are also introduced numerically, and the geometric properties of it are likely to be glossed over.

As such, the only way I know to introduce the subject is mathematically... unfortunately I'm too busy to try and devise a nice synthetic introduction to it.

Guybrush Threepwood
you've got me. I should have said..
"induce" is not the most correct word
mes excuses...(I hope I wrote that correctly)

Originally posted by Hurkyl
...studying special relativity and space-time diagrams, but those are also introduced numerically, and the geometric properties of it are likely to be glossed over.

As such, the only way I know to introduce the subject is mathematically... unfortunately I'm too busy to try and devise a nice synthetic introduction to it.

I already know SR, space-time diagrams, QM, ...

My interrogation is: What are the physical consequences or interpretations onto other physical "properties" while inserting an imaginary part to Ds.

For example:
Originally posted by Tyger
"None of the physical quantities are imaginary"
That's a really BIG assumption.
Could Hurkyl's reply be interpreted as
Would it mean that physicists are able to "observe" only squared distance (in 4D) and we use unsquared distance to solve toward whatever and therefore introduce root(-1)?
Would it mean that physicists are able to "observe" only squared things (in 4D)?
Originally posted by Fando
"...something that comes with framing things a certain way"
So ...
Originally posted by Fando
"In other words, in flat spacetime, using ict is sensible, but it can't really be used in a physics of curved spacetime"

P.S.: IMHO, mathematics are there to express physical thoughs, a common language to express ourself and to share ideas.

octol
My interrogation is: What are the physical consequences or interpretations onto other physical "properties" while inserting an imaginary part to Ds.
IMHO, since ds can be (and was originally) expressed without ict the imaginary part doesn't really need any physical interpretation other than it is just a mathematical way of expressing the same thing in different way.

There are no physical consequences when "inserting an imaginary part to Ds." as nothing is inserted, just reformulated.

Originally posted by octol
IMHO, since ds can be (and was originally) expressed without ict the imaginary part doesn't really need any physical interpretation other than it is just a mathematical way of expressing the same thing in different way.

There are no physical consequences when "inserting an imaginary part to Ds." as nothing is inserted, just reformulated.

May be "adding" instead of "inserting", depending on the point-of-view? Adding a dimension, ict, to three dimensions, x, y, z?

Since in ds2 = (c dt)2 - (dx2+dy2+dz), terms don't have the same sign, is this involve something larger than just "expressing the same thing in different way."?

Do terms have different signs?

octol
Since in ds2 = (c dt)2 - (dx2+dy2+dz), terms don't have the same sign, is this involve something larger than just "expressing the same thing in different way."?
Hmm doesn't the different signs just come from

speed = position / time
c = dr/dt
c^2 = (dr/dt)^2
c^2 = (dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2
(c dt)^2 * dt^2 = dx^2 + dy^2 + dz^2
(c dt)^2 - (dx^2 + dy^2 + dz^2) = 0
which is defined to spacetime interval ds^2

Staff Emeritus
Gold Member
My interrogation is: What are the physical consequences or interpretations onto other physical "properties" while inserting an imaginary part to Ds.

d&tau; and dl are observables. ds is how we compute them. i is a tool used in the computation. (absolute value could be used instead of i)

The point to what I'm trying to bring out is that the geometric studies gives a sense of why things should be the way they are... of course it all stems from the constant speed of light.

fando
Fando:
1) "in flat spacetime, using ict is sensible"! Would you explain "sensible"? (excuse my english)

Hmm, that line needs a little more explanation. What I meant to say was that the math that gives rise to "ict" can only work in Euclidean, or flat, spacetime. It won't work in curved spacetime. For that, we need differential geometry, and we don't need no "ict" for that.

Thanks