Geometry of space and Relativity

The geometry of Euclidean space and types of space in Einstein's Theory of Relativity. Reference to the geometry of space of Minkowski and Riemann.

Cover of PDF Theory of Relativity, Elements, and Criticism. Illustration of sailboat on purple sea.



Author: José Tiberius

Technical assistant:
Susan Sedge, Physics PhD from QMUL



2.c.2.b) Spatial geometry

This section will attempt to emphasize upon the difficulty the brain has to reason when there is so much terminological variety involved. On occasions, rather than talking about errors or mathematical curiosities, one would have to talk about mental eccentricities. Let us first go over the concepts of space from the spatial geometry of physics:

  1. Euclidean geometry of space

    • Normal Euclidean space

      Euclid’s spatial geometry is a mathematical abstraction that configures a space with the three dimensions that we observe with our eyesight or sense of touch. Because of the abstract nature of Euclidean geometry, space is fixed and absolute. That is to say, if correctly defined, it would become unalterable, as abstract space is independent of its contents.

      In other words, in Euclidean space, when an object becomes bigger, space remains unchanged.

      The terms contraction and expansion of space are meaningless in Euclidean spatial geometry.

    • Spatial localization and its perception

      The localization of objects in the Euclidean geometry of space is independent of the mechanisms used in its determination. However, our own eyes, as well as any other instrument may make mistakes, and have a limiting level of precision.

      We could mention here any mirror effect or similar to this, including the magnifying effect of light as it passes close to stars or gravitational lens effects. This difference between real localization and its information does not alter the abstract, absolute, and objective nature of space as a property assigned to physical objects.

    • Optical effect of the ordinary observer

      This effect appears with distance; we all know that distant objects look smaller, at least in regular or Euclidean spatial geometry.

    • Optical effect due to the speed of light

      Continuing with visual appearance, in 1959 there was an analysis of the presence that objects would have when they were in rapid movement, because of the effect of the small temporal difference in the perception of light coming from the part of the object closest or furthest from the observer.

      As it was discussed, the object appears longer than its actual size, as the rays of light that reach our eyes simultaneously correspond to two different moments. The ray of light coming from the part furthest from the observer is older than the closer other. Consequently, as the object is in motion, there will be a small difference between reality and its perception.

      The previous visions are produced in a Euclidean spatial geometry and must not be confused with those expressions that say that space curves, gets smaller, is contracted, etc., which are a consequence of Einstein’s Theory of Relativity and which we will mention further on.

  2. Geometry of love

    The geometry of subjective space, the geometry of love or life is very variable, so variable that at times, just like time, we do not perceive it; the example of being asleep is sufficiently clear.

    Another manifestation of subjective geometry would be the one mentioned when talking about the perception of the space-time of a bubble in the book The Equation of Love.

    Geometry of color of love
    Abstract square about time color geometry in a purple sea.

    There are other geometries of love, which are non-mathematical or purely spatial, but it would be better not to talk about them in this section.

  3. Relativistic spatial  or space-time geometry

    • The Lorentz-Fitzgerald contraction in the direction of movement

      The Lorentz transformations work with space in a similar way to the way I explained they do with time. One adds a fourth axis to the geometry of Euclidean space and the three typically spatial dimensions.

      The consequence of the geometry of space of this relativistic variant is that one object will be of different sizes for different observers. It is not that they will appear to have different sizes (we all know that objects at a distance look smaller than closer). It is that their sizes are truly and simultaneously different. Of course, one would also have to say what simultaneity is; like time, because it is relative, is also altered for one same temporal abstract moment.

      I think it is more to do with a change in the measurement units of each observer; as the reality should be unique. If indeed, it exists, of course!

    • Einstein’s Special Relativity

      This concept is identical to the previous one, except that it does not say whether things are bigger or smaller. It simply states that it is space that contracts or expands, according to the observers. It is the Hermann Minkowski space-time.

      Indeed, the relativity of space does not add anything new to the consistency or inconsistency of the dilation of time in the Theory of Special Relativity, except that it seems that a meter is a lot shorter than what it is for a meson. This particle travels 600 meters before disintegrating, but from the surface of the Earth, a relativist observer would swear it was 9500 meters.

      Something quite fun about the relativistic spatial geometry is that, despite the speed of light being constant, the objective space traveled in a second is not always the same. As a second is relative, and the meter is distant covered by the light in one second, by relativistic definition, the light will go almost 300 million meters in one second; when the second is shorter, the meters will be smaller too.

    • Geometry of space in General Relativity

      Let us fast-forward a bit; if Einstein’s Theory of Special Relativity dilates and contracts space, when one adds the axis of time to the three spatial Euclidean dimensions, the General Theory of Relativity –also Einstein’s– will curve these axes according to gravity. We could mention the developments or comments by Stephen Hawkins and Roger Penrose from the 70s onwards. Likewise, the so-called Riemann geometry and Schwarzschild metrics may produce tensions in many dimensions.

      This spatial geometry is difficult to explain because when they say space does not dilate but is the distance between two points in space that becomes larger, we end up losing myself entirely due to the lack of vocabulary for so many space-conceptual relativities.

      We have attempted to understand what it could mean that space or its geometry dilates. Maybe it is referring to –amongst other things– the fact that if light, as it travels on the gravitational field, curves independently of the gravitational attraction; then one could think that it is space that has changed. We would not consider it the most appropriate interpretation, but at least it would make some sense.

      It would be more precise to say that when light travels on the gravity field –tension of the radial symmetry of the reticular structure of matter–, the energy exchange produces a curvature effect on the propagation of light concerning the Euclidean space. The book Physics and Global Dynamics explains the cause of the Merlin effect, which is no more than a small gravitational force in addition to Newton’s force.

      A different topic is that of drag; let us imagine a Vinyl disc spinning on a turntable. If we place an object upon this disc, the object will turn –but not because of the effect of gravity, but because it is being dragged by the disc. Although traditional gravitational forces cannot explain it, and although it could be correct to a certain extent, we would not call this a geometric effect of the curvature of space-time-disc. We would just call it the drag of the Vinyl-Disc experiment.

      A different topic is that of drag; let us imagine a music disc spinning on a turntable. If we place an object upon this disc, the object will rotate –but not because of the effect of gravity, but because it is being dragged by the disc. Although traditional gravitational forces cannot explain it, and although it could be correct to a certain extent, I would not call this a geometric effect of the curvature of space-time-disc. I would simply call it the drag of the Vinyl-Disc experiment.

  4. Geometry of quantum space

    There is a tendency in Quantum Mechanics to deny the existence of space, as we understand it. The idea is to reduce the geometry of space to a set of discrete points and turn it into an analytical geometry in three dimensions –or however many are necessary to represent the experimental observations with the particular mathematical model in use.

    It is a substantial problem, probably of a sociological nature, to confuse mathematical dimensions with physical ones. For some people, any mathematical variable could be an additional spatial dimension. I would say that one should be evident to the fact that spatial dimensions are very different to many other variables, even if a computer does not quite differentiate between one and another.

  5. Spatial geometry in String Theory

    With this geometry of space, we could spend our time playing hide-and-seek, because with so many dimensions it cannot be easy to find adequate concepts to describe the physical reality. It seems that it deals with an intensive use of mathematics.

Of the five points we have mentioned about ways of understanding the geometry of space, we believe the first two coexist, while the last three are more or less recognized theories, but that cannot contribute direct experiments, due to the very abstract nature of space and obvious physical reality.


We will now attempt to explain the physical meaning behind some geometries of space, not necessarily in an academic fashion.

  • Flat geometry of Euclidean space

    Let us do magic; let us try to define tridimensional Euclidean space using an element of two-dimensional flat geometry solely.

    Recalling Plato the Greek, we could make the following definition of the geometry of space in three dimensions: “It will be the three-dimensional space, which will project shadows upon a two-dimensional plane, by the so-called laws of the sunshade.

    Another example would be the projections of three-dimensional harmonic waves upon a plane or element of plane geometry. Do not be scared; a good enough approximation would be to imagine the shadows of two balls bouncing on a sunny day.

    The same would happen for an analytical geometry of three dimensions or Euclidean geometry. Of course, it has a trick answer, just like all good magic: the third dimension is not included in the two-dimensional Euclidean space of reference, but in the equations that would express the laws of the sunshade, which indeed transform it into an analytical geometry of three dimensions.

    It is interesting to note that the equations of the aforementioned little laws would contain information about a world much more complicated than the two-dimensional world of reference. Because of this, they would have a much more general application than those laws, which describe a two-dimensional Euclidean space or flat geometry.

    In other words, one cannot define a Euclidean space or plane that folds over or allows other magic tricks, because it would be playing with the language.

    A third dimension can be “folded” and integrated or superimposed on a flat geometry, but the two dimensions of the plane will remain unaltered, or at least with the same rules they had unless we change them too. In this case, we would be breaking the plane, the train, the concept, and everything.

    It would be too much like the theorem of the fat point, by which two parallel lines pass through.

    We should highlight that including a new type of relation which affects the reference coordinates or axes of the plane is equivalent to adding new dimensions, where these would be the laws which govern their change or variation. It is a fundamental concept of geometry and mathematics.

    In fact, this is what I think the Lorentz transformations do with their equations.

    It may have been convenient to search for equations with more variables, which would facilitate specific calculations and some comparisons, in the same way, that Relativity undoubtedly does. However, these must not make one lose the notion of fundamental physical concepts for the logic of our nature, such as objective time and space.