Metric, Minkowski-metric, Minkowski-space

Idea: This chapter provides an interpretation of the Minkowski metric and hence of all special relativistic effects.

Thesis

In relativistic physics, space and time are unified by introducing a four-dimensional spacetime. However, time still plays a special role:

Humans experience, observe and measure time in a fundamentally different way than they do with space: one can return to a place already visited, yet may not return to an earlier point in time. Similarly, one may, from a certain position, look onto another place (“overlook space”), yet the only point in time observable is one single moment.

This basic observation shall be translated into a mathematical model, starting with the following two definitions:

Definition 1:
A universal observer has a complete overview of all dimensions - even time.

Definition 2:
A flat observer, or “single-layered” observer, does not overlook all dimensions completely, but sees only a single point of one or more dimensions.

For example, a flat observer in a three-dimensional Euclidean space with coordinates x, y and z who has a “flattened” view of the z-axis, would see only the x and y dimensions entirely, whereas he would see the z dimension only at a single point, e.g. z = 4. He would therefore see a plane and not the entire 3d-space.

Corollary:
The consequence for a flat observer is that, in the dimension for which he can see only one single point at once, he is not able to define a scale for direct measurement. For, to be able to do so, he would have to see at least two points of said dimension.

If there is no movement within this dimension, then it is indifferent, for the dimension is not even perceived. If, however there is movement within this dimension, the flat observer perceives it, but he cannot provide it with a scale directly.

Thus, his only option is it to define an indirect scale: By transferring the perceived movement to already known scales; so, in the xyz case above, by mapping the perceived movements in the z-direction to a scale which lies in the xy-plane.

Observation 1:
Humans are, according to definition 2, flat observers with respect to the dimension of time:

Independently of what we do, our measurement and perception of time is always reduced to a single point on the time axis. Therefore, humans will never be able to perceive or measure a time interval directly. The measurement of the time scale always happens indirectly, by measuring movement in a spatial dimension and mapping it on the time scale.

A universal observer is theoretically possible, but does not exist in the physical world.

As an example/ illustration 1, let us consider the light clock:

Two mirrors are set up at a certain distance from each other. A light ray travels back and forth between the two mirrors. Every time the light ray reaches a mirror, it corresponds to the passing of an interval in the time scale.

The time scale is therefore measured by the distance v between two objects. The distance between the two mirrors, in abstract form, corresponds to a vector.

This example illustrates a general principle, which can be formulated as follows:

Observation 2:
An indirect measurement of time by a flat observer is always conducted using experimental setups in the directly measurable dimensions.

In all experiments, one deals with real arrangements, which can be described as vectorial quantities (like the distance v in the example above). In an Euclidean space, vectorial quantities transform contravariantly under coordinate transformations. Hence, each experiment has contravariant transformation properties (for a definition of contravariant, see e.g. ^[1], for short introductions e.g. ^[2] & ^[3]). This results in the following corollary:

Corollary:
Because of the indirect measurement of the time scale via experiment, the contravariant transformation property of the experiment is transferred to the time scale. The applied basis component in the time direction therefore transforms contravariantly.

Since the basis of Euclidean space transforms covariantly, this means that the flat observer chooses a basis of spacetime in which the basis vectors transform covariantly in three directly measured dimensions (space), but contravariantly in one indirectly measured dimension (time). Thus, the question must be raised, on how the flat observer can reach a unified mathematical description when using this mixed basis.

To answer this, the following thesis is proposed:

Thesis:

The common treatment of basis vectors, whose components transform partly covariantly, partly contravariantly, leads to vector spaces with negatively signed components in the metric tensor.

Thus, the indirect measurement of a direction by the flat observer defines a mapping from the Euclidean- into the relativistic Minkowski-space.

Proof

The most recent version of this paragraph is found in article 1 [REF]

The four-dimensional Euclidean space with orthonormal basis $\mathbf {e}$ is observed. Due to the normalization, basis transformations are length conserving.

The possible linear, length conserving coordinate transformations in a four dimensional Euclidean space are equivalent to the Lie-group $SO(4)$ , which can be represented by the special orthogonal matrices of order four. These matrices are orthogonal and have a determinant equal one. (For a short introduction see ^[4] & ^[5], for a more complete treatment see e.g. ^[6] (online) & ^[7]).

As a consequence of the orthogonality, the inverse of the matrices is simply its transpose, i.e. $\mathbf {U^{-1}=U^{T}}$ for all $\mathbf {U}$ in $SO(4)$ .

Let’s consider a single transformation matrix. It corresponds to an active rotation of a vector’s coordinates from $\mathbf {v}$ to $\mathbf {\overset {\sim }{v}}$ , with respect to a basis $\mathbf {e}$ :

\mathbf {e^{T}{\overset {\sim }{v}}=e^{T}\left(Uv\right)}

</equation>

On the other hand, it could also be interpreted as an active rotation of the basis, if the transformation matrix is applied to the left side:

\mathbf {{{\overset {\sim }{e}}^{T}}v=\left(e^{T}U\right)v={\left(U^{T}e\right)}^{T}v={\left(U^{-1}e\right)}^{T}v}

</equation>

It is apparent that the transformation on the coordinates of a vector is equivalent to the inverse transformation on the basis. Alternatively, by interpreting the coordinates of a vector as a basis, one could rotate it in the dual space.

The important thing to note is, that active transformations on vector coordinates or on basis components are not equivalent, but in fact inverse to each other.

If now a contravariant component is added to the covariant basis it transforms inversely to the others and the basis cannot be interpreted as a tensor basis of Euclidean space.

Thus, the question arises, if there is a vector space in which basis and vector elements can be interpreted as tensors again. To do this, we will take a closer look at the transformations. They can be expressed as an exponential series:

\mathbf {U} =e^{t\mathbf {A} }\qquad \qquad \qquad \qquad \mathbf {U^{-1}} =\mathbf {U^{T}} =e^{-t\mathbf {A} }

</equation>

Where $t$ is the parameter of the transformation (the angle of rotation) and $\mathbf {A}$ is a skew symmetric matrix, i.e. $\mathbf {A^{T}=-A}$ . To be more precise the matrices $\mathbf {A}$ build the Lie-Algebra $so(4)$ of $SO(4)$ : This is the algebra that generates the infinitesimal linear coordinate transformations which span the entire Lie-group $SO(4)$ .

By looking at infinitesimal transformations (very small $t$ ), it is sufficient to consider only the linear terms of the exponential series:

\mathbf {U} =e^{t\mathbf {A} }\approx {}(\mathbf {1} +t\mathbf {A} )\qquad \qquad \qquad \qquad \mathbf {U^{-1}} =e^{-t\mathbf {A} }\approx {}(1-t\mathbf {A} )

</equation>

Here, the special role of the Lie-algebra becomes apparent. By applying the transformation on the basis on the left side (elementwise notation on the right):

\mathbf {{\overset {\sim }{e}}^{T}} =\mathbf {e^{T}U} \approx {}\mathbf {e^{T}} (\mathbf {1} +t\mathbf {A} )\qquad \qquad \qquad {\overset {\sim }{e}}_{l}={e}_{k}U_{\ \ \ l}^{k}\approx {}{e}_{k}{(1+tA)}_{\ \ \ l}^{k}

</equation>

By reorganizing the terms one gets the change of the basis through the transformation:

\mathbf {{de}^{T}} =\mathbf {{\overset {\sim }{e}}^{T}-e^{T}} =t\,\mathbf {e^{T}A} \qquad \qquad \qquad {de}_{l}={\overset {\sim }{e}}_{l}-{e}_{l}=t\,e_{k}A_{\ \ \ l}^{k}

</equation>

On the other hand, an active, infinitesimal transformation on the vector coordinates has the following effect:

\mathbf {\overset {\sim }{v}} =\mathbf {Uv} \approx {}(\mathbf {1} +t\mathbf {A} )v\qquad \qquad \qquad \qquad {\overset {\sim }{v}}^{\ j}=U_{\ \ \ i}^{j}v^{\ i}\approx {}{(1+tA)}_{\ \ i}^{j}v^{\ i}

</equation>

Thus, the vector changes by the following amount after the transformation:

\mathbf {dv} =\mathbf {{\overset {\sim }{v}}-v} =t\mathbf {Av} \qquad \qquad \qquad \qquad {dv}^{\ j}={\overset {\sim }{v}}^{\ j}-v^{\ j}=tA_{\ \ i}^{j}v^{\ i}

</equation>

To compare the change of the coordinates ${dv}^{\ j}$ to the change of the basis ${de}_{l}$ , the coordinates must be transferred from the vector space to the dual space. In Euclidean space this is achieved by transposition:

\mathbf {{dv}^{T}} ={(t\mathbf {Av} )}^{T}=t\,\mathbf {v^{T}A^{T}} =-t\,\mathbf {v^{T}A} \qquad \qquad {dv}_{j}=t\,v_{i}A_{j}^{\ \ \ i}=-t\,{v_{i}A}_{\ \ j}^{i}\qquad \left(\not =te_{k}A_{\ \ \ l}^{k}\right)

</equation>

In the last step, the skew symmetry of $\mathbf {A}$ was used $\left(A_{l}^{\ \ k}=-A_{\ \ \ l}^{k}\right)$ . As expected, when comparing with the earlier equation one notices that this does not correspond with the transformation rules of a basis.

If we leave the Euclidean space, the change from vector space to dual space must be done by use of the metric tensor^[8]. We assume that it has a diagonal form:

g_{ij}=g\cdot \delta _{ij}

</equation>

The transformation rule for the vector components into the dual space is then:

{dv}_{l}=g_{lj}{dv}^{j}=\left(g_{lj}g^{ik}tA_{\ \ i}^{j}g_{ki}v^{i}\right)=-g\cdot t\;{v_{k}A}_{\ \ \ l}^{k}\qquad \left(=:\ t\;e_{k}A_{\ \ \ l}^{k}\right)

</equation>

The comparison with the change of the elements of a basis vector (rightmost term) makes it visible that for the vector to adopt the same transformation properties as the basis, the corresponding element of the metric must be $g_{lj}=g=-1$ . Now the change of the basis element constructed from the vector corresponds to the change of a regular basis element:

{dv}_{l}=-g\cdot t\;{v_{k}A}_{\ \ \ l}^{k}=t\;{v_{k}A}_{\ \ \ l}^{k}={de}_{l}

</equation>

Thus, it was shown: If a vector which transforms contravariantly in Euclidean space is used as a basis element, a vector space can be constructed where all basis elements transform covariantly again. This vector space has a corresponding element in the metric with a negative entry. $\square$

Implications

The new metric enables directly the definition of an adjusted scalar product and therefore a norm. Though this norm is not positive definite.
The newly found mixed space is again a vector space. It fulfils all the properties of a vector space with exception of the positive definite norm, and is therefore a Minkowski vector space.
Particularly, it follows that the negative elements of the metric tensor observed in relativistic physics are an artefact of the measurement done by a flat observer – and are not an intrinsic property of spacetime. The uncurved spacetime can be understood in terms of the traditional four-dimensional Euclidean space, where the physics takes place, and only through the measurement by a flat observer the negative metric elements arise.

For further illustration

Example 2 – normal and mixed rotation
First, we consider active rotations in a traditional Euclidean vector space. One can either rotate the vectors (contravariant rotation), or – equivalently to this – rotate the basis in the opposite direction (covariant rotation):

Remarks:

In both cases, the length of the vector ds is conserved, in accord with the Euclidean norm.
Please note the lower indices of the basis vectors, which represent their covariant transformation.

If we now consider a space in which one of the basis vectors transforms contravariantly – that is, as we would expect a vector to – the following happens:

In a contravariant rotation, which in fact should rotate the vectors only, the basis vector rotates as well; for it transforms contravariantly, too. In the diagram, the basis vector is indicated accordingly with an upper index. The other, covariant basis vectors remain unchanged instead:

After the rotation, distances in the y-direction are no longer measured along the original y-axis, but parallel to the rotated basis vector $\mathbf {e^{y}}$ . This direction is still perpendicular to the vector ds, (red angle icon, on the right), since both objects were rotated by the same angle.

As a result, the Pythagorean triangle, used for the distance measurement, has been changing in comparison to the earlier diagram: dx’ is the new hypotenuse of the triangle and the sign is inverted (red).

Open questions

Only the mathematical thesis can be proven. The remaining arguments are observations. Can they be disproved, or can they be supported further?
How could one disprove the observation that one can only measure one point in time? Is there no experiment, which measures two points in time?
The mathematical treatment was only performed in uncurved spaces. Could further effects arise in the general case?
Similarly, the mathematical treatment was only performed for linear transformations. Is a treatment for affine transformations also needed? What would it look like?
As a computational trick a similar relationship between Euclidean and Minkowski space is already known, the Wick-rotation (complexification of the time coordinate). What is the exact connection to the procedure presented here?

References

↑ Contravariant Tensor, Wolfram Mathworld, accessed 23.5.2017 [1]
↑ Elektrodynamik, G. Graf, pp. 97ff, Anhang C: Tensorkalkül, ETH Zürich, 2008 [2]
↑ Seminar I: Vektoranalysis, Krummlinige Koordinatensysteme, Differentialoperatoren und Integralsätze, D. Hansen, FSU Jena, Theoretisch Physikalisches Institut, 2002 [3]
↑ Lie-Gruppen und Lie-Algebren Eine Einführung, D. Schliebner, Humboldt Universität zu Berlin, 2009 [4]
↑ Theoretische Physik, M. Bartelmann, B. Feuerbacher, T. Krüger, D. Lüst, A. Rebhan und A. Wipf, in Kapitel 27, Springer, 2014, p. 872ff.
↑ An Elementary Introduction to Groups and Representations, B. C. Hall, arXiv, 31.5.2000 [5]
↑ Lie Groups, Lie Algebras, and Representations, B. C. Hall, Springer, 2003.
↑ Metric Tensor, Wolfram Mathworld, accessed 19.5.2017 [6]

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[Wolfram1-1] Contravariant Tensor, Wolfram Mathworld, accessed 23.5.2017 [1]

[Graf-2] Elektrodynamik, G. Graf, pp. 97ff, Anhang C: Tensorkalkül, ETH Zürich, 2008 [2]

[Hansen-3] Seminar I: Vektoranalysis, Krummlinige Koordinatensysteme, Differentialoperatoren und Integralsätze, D. Hansen, FSU Jena, Theoretisch Physikalisches Institut, 2002 [3]

[Schliebner-4] Lie-Gruppen und Lie-Algebren Eine Einführung, D. Schliebner, Humboldt Universität zu Berlin, 2009 [4]

[Bartelmann-5] Theoretische Physik, M. Bartelmann, B. Feuerbacher, T. Krüger, D. Lüst, A. Rebhan und A. Wipf, in Kapitel 27, Springer, 2014, p. 872ff.

[Hall1-6] An Elementary Introduction to Groups and Representations, B. C. Hall, arXiv, 31.5.2000 [5]

[Hall2-7] Lie Groups, Lie Algebras, and Representations, B. C. Hall, Springer, 2003.

[Wolfram2-8] Metric Tensor, Wolfram Mathworld, accessed 19.5.2017 [6]

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Metric and time measurement

Contents

Thesis

Proof

Implications

For further illustration

Open questions

References

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Metric and time measurement

Thesis

Proof

Implications

For further illustration

Open questions

References

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