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Using homogeneity of space, isotropy of space and the principle of relativity (without the constancy of light speed), one can derive:

$$x' = \frac{x-vt}{\sqrt{1+\kappa v^2}}$$

$$t' = \frac{t+\kappa vx}{\sqrt{1+\kappa v^2}}$$

$\kappa = 0$ denotes Galilean and $\kappa < 0$ denotes Lorentz Transformation.

What does $\kappa > 0$ denote? Is it physically possible? I was told that it is self-inconsistent. Can somebody help me with a proof of this?

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    $\begingroup$ Isn't this basically the One Big Lie of the setting of Greg Egan's Orthogonal series? gregegan.net/ORTHOGONAL/ORTHOGONAL.html#CC $\endgroup$ Aug 5 '20 at 19:07
  • $\begingroup$ That's a really interesting concept for a novel series. I had never heard of Orthogonal before. $\endgroup$ Aug 5 '20 at 19:16
  • $\begingroup$ It reminds me a bit of the approach to world building of Tolkien (himself a language professor). I am intrigued . . . . $\endgroup$
    – m4r35n357
    Sep 18 '20 at 11:33
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$\kappa > 0$ represents Euclidean geometry, in which the time axis is equivalent to (and freely interchangeable with) the spatial ones. In other words it acts like a fourth spatial dimension.

So you can for example take a left turn into the time axis and go forward, then turn around and go back in time. Many consider this non-physical, leaving the choice beweeen $\kappa = 0$ (Galilean Transformation) and $\kappa < 0$ (Lorentz Transformation).

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    $\begingroup$ you can do the same in Minkowski and I think also in "Galiliean space". Time travel is forbidden by additional structure introduced there. I do not see any reason why some causal structure cannot be introduced in Euclidean geometry also. $\endgroup$
    – Umaxo
    Aug 5 '20 at 8:56
  • $\begingroup$ But the "structure" introduced in Minkowski space (the null interval) is only there because of the value of $\kappa$, so I think your comment is a tautology. Also, in "Galilean space" and "Galilean time" there is no linkage at all between time and space, so I'd be interested to hear how it could be done. $\endgroup$
    – m4r35n357
    Aug 5 '20 at 10:18
  • $\begingroup$ Sure, the causal structure in both of them is introduced quite naturally. But you can draw time traveling loop in both spaces just fine, and there is nothing in the geometry itself that forbids material particles traveling on them. It is separate and additional law, albeit naturally introduced. In case of Euclidean geometry, it would need to be supplied by hand. Just introduce some vector field and demand that causal chronology is given by sign of the result of contraction of metric on tangent vector with this field and forbid the travel on curves which produce, say, negative sign $\endgroup$
    – Umaxo
    Aug 5 '20 at 10:30
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As said in the answer of @m4r35n357 it is the case of Euclidean geometry. To see this, look at the transformations, that preserve the distance : $$ds^2 = dx^2 + dt^2$$

Among with the translations there are also rotations: $$ \begin{pmatrix} t^{'} \\ x^{'} \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} t \\ x \end{pmatrix} $$ Look, for example, at the first row. After defining $v = \tan \theta$, it is actually the $t$ transformation, with $\kappa = 1$: $$ t^{'} = \frac{t + v x}{\sqrt{1 + v^2}} $$ This geometry is obtained from the Minkowski space by Wick rotation $t \rightarrow i t$. It is mathematically consistent and alright, however, our world is described by the metric (in the flat case), with the minus sign between $dx^2$ and $dt^2$.

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  • $\begingroup$ So there is no inherent inconsistency? $\endgroup$ Aug 5 '20 at 9:00
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    $\begingroup$ @PhyEnthusiast, I would say so, the main reason to set $\kappa > 0$ is that nature is created in a such way, the waves propagate, according to the D'Alambert equation (classically), not the Laplace $\endgroup$ Aug 5 '20 at 9:09
  • $\begingroup$ Is it even possible for waves to propogate, if they follow laplace equation instead of d'Alembert? $\endgroup$ Aug 5 '20 at 9:16
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    $\begingroup$ @PhyEnthusiast, yes, It wouldn't be the wave in ordinary sense, some solution, decaying with a power law in $D >= 3$ (logarithmically increasing in 2D). . $\endgroup$ Aug 5 '20 at 9:19
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    $\begingroup$ The science fiction author Greg Egan wrote the Orthogonal trilogy about a universe with this metric. Highly recommended. $\endgroup$ Aug 5 '20 at 20:17
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As stated in almost all the answers, $\kappa>0$ does indeed define a set of transformation that preserves the Euclidean metric, with the Lorentz group $SO(1,3)$ being replaced by $SO(4)$. Although this does make mathematical sense, there's an internal physical inconsistency, which shows that an universe with $\kappa>0$ is not possible.

To show the inconsistency, first note that homogeneity and isotropy of space along with the principle of relativity not only gives us the spacetime transformations, but also gives us the velocity addition rule$^*$, which looks like $$w=\frac{u+v}{1-\kappa uv} .$$ Since $\kappa>0$, assume $\kappa=1/c^2,\ c\in\mathbb R,\ c<\infty$. Then $$w=\frac{u+v}{1- uv/c^2}.$$ We first show that there exists a velocity greater than $c$. To show this, consider $u=c/2=v$. Then $w=c/(3/4)=4c/3>c$.

Now, let $\gamma_u=1/\sqrt{1+\kappa v^2}=1/\sqrt{1+v^2/c^2}$. The spacetime transformation tells us that $\gamma_0=1$ and hence the square root in $\gamma_u$ is the positive square root and thus $\gamma_u>0\ \forall u\in \mathbb R$.

The assumed postulates also lets us derive$^*$ $$\gamma_w=\gamma_u\gamma_v(1-\kappa uv)=\gamma_u\gamma_v(1-uv/c^2).$$

Consider $u,v$ such that $uv>c^2$. This is possible because there exist velocities greater than $c$ as we have shown. Then $1-uv/c^2<0\Rightarrow \gamma_w=\gamma_u\gamma_v(1-uv/c^2)<0\Rightarrow \gamma_w<0$ which is a contradiction.

This is the internal inconsistency. The postulates do not allow for an universe with $\kappa>0$.

$\rule{20cm}{0.4pt}$

$^*$ The derivation of these facts is very beautifully shown in this paper. It's simple and concise, and the proof of inconsistency is also in the paper.

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There has been much discussion on one-way speeds of light and simultaneity etc. in the philosophy of physics literature. Most famously, Reichenbach introduced a parameter $\epsilon$, which gives the (one-way) light speed in opposite directions as $c/2\epsilon$ and $c/2(1-\epsilon)$. Here $c$ is the "two-way" speed of light, which can actually be experimentally measured.

One way of interpreting this discussion is as ordinary relativity but described on a "tilted" hyperplane: one which is not orthogonal to the observer's 4-velocity. This is the approach of papers like Ungar 1991, see equation 9 for the one-way Lorentz transformations. I haven't analysed your $\kappa$ parameter specifically. But it is certainly consistent to describe relativity using coordinates that are tilted relative to given observer(s).

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