B Imaginary distance is nonsense

[AndreyN]

TL;DR Summary

Imaginary distance contradicts to the axiom of space

If we depict any real number by corresponding segment of strait line then its necessary to correspond two segments to both real and imaginary parts of complex number. Both of segments have to lay on the vector with imaginary length in minkowski space which situation contradicts to the axiom of space "For every two points there exists no more than one line that contains them both".
Therefore imaginary distances in minkowski space are nonsense.
What is wrong? Explain please.

 

[Dale]

TL;DR Summary: Imaginary distance contradicts to space axiom

the axiom of space "For every two points there exists no more than one line that contains them both".

1) That is not an axiom of space
2) It is also not an axiom of Euclidean space
3) It is also not an axiom of Minkowski spacetime

Therefore imaginary distances in minkowski space are nonsense.
What is wrong?

1) Your premises are wrong
2) Your reasoning from those premises is not sound
3) Your conclusion doesn’t follow from your reasoning

 

[Nugatory]

TL;DR Summary: Imaginary distance contradicts to space axiom

What is wrong? Explain please.

There are no imaginary distances in Minkowski space.
The sign of the squared interval tells us (by convention) whether the interval represents a proper time or proper distance. If we have chosen our sign convention such that the squared norm of a spacelike interval is negative then we drop the negative sign before taking the square root to get a distance, just as we would to calculate a proper time if we were using the other sign convention.

Note also that unlike proper time, a distance calculated in this manner is of less physical interest: it's the distance between the endpoints of a ruler at rest in a particular inertial frame, not especially useful except when doing calculations using that frame.

 

[AndreyN]

$e^2=-1, e=?$

 

[Nugatory]

$e^2=-1, e=?$

$e=i$. So?
Your question is more relevant if we're using the obsolete $ict$ formalism to describe Minkowski space, but that's just another argument against using that formalism (which fails dismally in curved spacetimes, so is a dead end for studying relativity).

 

[AndreyN]

Regarding the Minkowski space, in the work of L. D. Landau and E. M. Lifshitz [L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Third Revised English Edition (Course of Theoretical Physics Volume 2) (Pergamon Press, New York, NY, 1971)] in section 2 on the page 4 it is stated the following

"$$ds^2=c^2dt^2−dx^2−dy^2−dz^2$$ (2.4)

The form of expression ... (2.4) permits us to regard the interval, from the formal point of view, as the distance between two points in the fictitious four-dimensional space (whose axis are labeled by x; y; z, and the product ct). But there is a basic difference between the rule for forming this quantity and the rule in ordinary geometry: In forming the square of the interval, the squares of the coordinate differences along with the different axes are summed, not with the same sign, but rather with varying signs».

If $ds^2=-1$ then ds=?

 

[AndreyN]

That is not an axiom of space

In https://en.m.wikipedia.org/wiki/Hilbert's_axioms
N 2.

 

[Nugatory]

In https://en.m.wikipedia.org/wiki/Hilbert's_axioms
N 2.

And even before we get #2 in that article we see "Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie[1][2][3][4] (tr. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff."

And you do understand that Minkowski space is not Euclidean?

You have been blocked from further posting in this thread because you are just repeating the same misunderstanding, which is a waste of your time and everyone else's.

We are leaving the thread open on the off chance that someone is aware of an axiomatization more appropriate for Minkowski space.

 

[Ibix]

We are leaving the thread open on the off chance that someone is aware of an axiomatization more appropriate for Minkowski space.

Ropey internet connection isn't letting me download the PDF to check or provide a decent reference, but from memory @bcrowell's GR book offers one.

 

[thomsj4]

Suppose we construct a real vector space endowed with the Minkowski bilinear form
$$
\langle u, v \rangle = u^0 v^0 - u^1 v^1 - u^2 v^2 - u^3 v^3,
$$
so that for two points $p_1$ and $p_2$, one defines
$$
\Delta s^2 = \langle p_2 - p_1, p_2 - p_1 \rangle.
$$
If $\Delta s^2 < 0$, one might write $\Delta s = i\sqrt{-\Delta s^2}$, but this $i$ is not a literal demand for a new “imaginary line segment.” Instead, it is just a symbolic shorthand indicating that the interval is spacelike with $\Delta s^2 < 0$. In fact, these two points $p_1$ and $p_2$ still lie on a single geodesic in Minkowski space, preserving the usual uniqueness property “for every two points there is exactly one straight line through them.” The error arises when one interprets the factor of $i$ as geometrically mandating a second segment perpendicular or disjoint from the first, as if real and imaginary parts had to be plotted on separate lines. But Minkowski space only needs one line between any pair of points; no second line is ever introduced. The negative sign in the metric simply encodes the causal structure (timelike vs. spacelike intervals), and $\Delta s = i\sqrt{-\Delta s^2}$ is an algebraic convention rather than a literal call for extra geometry. Hence there is no contradiction, and the “imaginary distance” does not invalidate the axiom that only one line passes through two distinct points.

 

[Dale]

In https://en.m.wikipedia.org/wiki/Hilbert's_axioms
N 2.

Space isn’t Euclidean. And neither is spacetime.

Edit: I see @Nugatory already made this point.

To expand on the point, no reasoning from the axioms of Euclidean geometry will say anything about relativity. The geometry of relativity is the geometry of a Lorentzian manifold. Lorentzian manifolds are well known to violate the axioms of Euclidean geometry. So the fact that you find such violations is utterly trivial. It certainly doesn’t imply any problem with relativity.

 

[Hornbein]

Spacetime is hyperbolic, not Euclidean/flat. That's why it's hard to get used to. In our everyday lives we don't come across hyperbolic spaces .

 

[pervect]

Spacetime is hyperbolic, not Euclidean/flat. That's why it's hard to get used to. In our everyday lives we don't come across hyperbolic spaces .

I'd have to disagree, though to some (but not all) of my disagreement is about word choices. It's fairly easy to make a few word substitutions in your response to get something I'd agree with.

Space-time in special relativity, i.e. Minkowskii space-time, is a flat, Lorentzian geometry. This viewpoint comes from Misner, Thorne, Wheeler's book "Gravitation", by the way. I'm not sure if anyone is curious enough to look up the reference, but the information is there for those who want a source or to read further discussion. It's one of my favorites, by the way, and I highly recommend it, even if it is quite old.

So the meat of my disagreement is where you imply that that Minkowskii space-time is curved. It's not - it's flat.

While I disagree with the specifics of your point, I agree with what I think is some of the underlying ideas. If I was re-writing your observation, I'd say that the geometry of space-time is Lorentzian (rather than hyperbolic), and get rid of the remarks about flatness.

A related point though, is that a Lorentzian geometry is not well-characterized as a geometry with a "complex distance". At least not in any formulation that I've ever seen.

A Lorentzian geometry is not "nonsense", but it's also not Euclidean. A Lorentzian geometry has the usual quadratic bilinear form that defines squared distance, but the quadratic form is not positive-definite, so the squared distance can be negative. This is different than saying that "distance is complex". So a geometry with a complex distance in this context is a bit of a straw man - I don't have any information about such a geometry, but I'd hesitate to say that it was necessarily inconsistent. It's mostly just not relevant to the geometry of Minkowskii space-time, which is Lorentzian.

Wiki's treatment of Lorentzian geoemetry, for those who might not have MTW's text, redirect to https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold#Lorentzian_manifold. It's not necessarily the best source on the topic, I was just searching for available information for those who want to read more and who may not have MTW's text.

 

[PeterDonis]

Euclidean/flat

These two aren't the same thing. "Euclidean" means a specific geometry. "Flat" just means the Riemann curvature tensor is zero. The latter is a weaker condition that is satisfied by other geometries besides the Euclidean one.

 

[PeterDonis]

Space isn’t Euclidean.

It can be if an appropriate coordinate chart on spacetime is chosen whose spacelike slices of constant time are Euclidean 3-spaces. (This does, AFAIK, place restrictions on the spacetime geometry, since I don't think all spacetime geometries permit such a chart.)

Whether such a chart, in spacetimes that permit it, has any physical relevance is a different question.

 

[Histspec]

Spacetime is hyperbolic, not Euclidean/flat. That's why it's hard to get used to. In our everyday lives we don't come across hyperbolic spaces .

When people refer to spacetime as hyperbolic, they usually mean that parts of spacetime can be described by certain models of hyperbolic space. For instance, relativistic velocity space inside $$u_{x}^{2}+u_{y}^{2}+u_{z}^{2}=1$$ can be described in terms of the Beltrami–Klein model or Poincaré disk model of hyperbolic geometry, while spacetime coordinates inside $$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=-1$$ can be described in terms of the hyperboloid model of hyperbolic geometry. All of those are special cases of the Cayley-Klein metric which is used to describe all sorts of (non)-Euclidean geometries.
Of course, none of this implies that spacetime as such is curved in the sense of Riemannian curvature.

 

[robphy]

When people refer to spacetime as hyperbolic, they usually mean that parts of spacetime can be described by certain models of hyperbolic space. For instance, relativistic velocity space

They shouldn’t refer to Minkowski spacetime as hyperbolic since they probably don’t refer to Euclidean space as circular or spherical or elliptic… for instance, the space of directions…

Minkowski spacetime is flat (zero Riemann tensor) but uses hyperbolic trigonometry,
just like Euclidean space is flat (zero Riemann tensor) but uses circular trigonometry.

Euclidean, Minkowski, and Galilean are zero Riemann curvature spaces… and are the affine Cayley-Klein geometries, where the parallel postulate applies. That’s why we can do vector algebra on these spaces.

From Yaglom’s “A simple noneuclidean geometry and its physical basis: an elementary account of Galilean geometry and the Galilean principle of relativity”

The last two rows are spacetime geometries.
The last row has a Lorentzian signature…
(1+1) deSitter, Minkowski, and anti-deSitter.

 

[Dale]

This does, AFAIK, place restrictions on the spacetime geometry, since I don't think all spacetime geometries permit such a chart.

Yes, which is why I said space isn’t Euclidean. My statement is correct in reference to the actual geometry of the universe

 

[PeterDonis]

which is why I said space isn’t Euclidean. My statement is correct in reference to the actual geometry of the universe

I'm not sure I understand. Our best current model of the universe is spatially flat. The spacelike 3-slices of constant FRW coordinate time (i.e., the time of comoving observers) are infinite Euclidean 3-spaces. That means this spacetime admits the kind of coordinate chart I described.

I understand that spacetime is not Euclidean. But you didn't say "spacetime". You said "space".

 

[Dale]

I'm not sure I understand. Our best current model of the universe is spatially flat. The spacelike 3-slices of constant FRW coordinate time (i.e., the time of comoving observers) are infinite Euclidean 3-spaces. That means this spacetime admits the kind of coordinate chart I described.

I understand that spacetime is not Euclidean. But you didn't say "spacetime". You said "space".

Yes. I said “space”. The universe also contains things like rotating and merging black holes and quasars and pulsars and so forth.

 

[PeterDonis]

Yes. I said “space”. The universe also contains things like rotating and merging black holes and quasars and pulsars and so forth.

I see, you are basically saying that once we take deviations from homogeneity and isotropy in our actual universe (as opposed to our best current cosmological model) into account, we cannot find a chart on the actual geometry with flat Euclidean 3-slices.

 

[Bosko]

TL;DR Summary: Imaginary distance contradicts to the axiom of space

There are important differences between the space-time of general relativity and the metric space in mathematics.
- Points in space-time are called events.
- "Metric" in that space does not satisfy all properties of the metric function in metric space.
For example: If the distance between two events is zero, it does not mean that it is the same event.
The distance between an event on Earth now and an event on the Sun in 8 minutes is zero.

Spacetime is a Riemannian manifold with a pseudo metric.
Spacetime is a pseudo Riemannian manifold.
A space of smooth differential geometry with a pseudo metric.