The 3D Space-Time Diagram
Its Discovery, Facts and Hypotheses
The Lorentz transformation of special relativity describes how measurements of space and time in a moving frame of reference depend on its motion relative to an observer in a stationary frame.
The first space-time diagram of the Lorentz transformation was revealed by Minkowski in 1908. Over the last century, this intuitive visual aid helped clarify Einstein's theory, facilitating its dissemination to scientists and the general public. Minkowski's diagram explains the space-time transformation with a two-dimensional drawing in which the space-time planes for the two frames are superposed. The coordinate axes for the moving frame P' rotate toward each other through an angle ψ that depends on the relative speed v between the two frames.
The Three-Dimensional Space-Time Diagram (3DSD), on the other hand, displays a 3D view of planes P and P'. Plane P' rotates through an angle δ into the dimension u, which until now has been missing from our view and understanding not only of special relativity but also of reality itself.
Comparing the two views, we see that Minkowski's diagram provides only a partial view of the 3D Space-Time Diagram. The rotation seen in Minkowski's diagram represents the projection of the axes of plane P' onto plane P. The relative speed v and angles δ and ψ vary from one drawing to another; the higher the speed, the larger the angles. (In the diagram, c denotes the speed of light.)
To understand how the 3DSD works, view the slide show found at the top of this page.1 The first slide provides a concise introduction to the 3DSD. The other four slides present the 3DSD in action, side by side with Minkowski's diagram, and show differing values of the relative speed between two frames of reference.
The second slide shows a pair of identical space-time frames represented by planes P and P'. The two frames are at rest relative to each other and appear to be indistinguishable, as if they were both in the same space-time. But that is not the case. There are two observers – one in each frame – and each observer can measure events not only in the frame where he is located, but in the other frame as well.
The observer in frame P measures the spatial and time coordinates of events "from the inside" in frame P, and "from the outside" those of events in frame P'. Those viewpoints are reversed for the observer in frame P'. In this slide, the measurements of events in either frame are identical for the two observers, because the two frames are at relative rest.
The other three slides depict scenarios in which frame P' is moving at different but constant velocities2 with respect to frame P. The plane of P' is shown rotated out of the space of frame P. Hence, the observer in P cannot make direct measurements of events in P'; that observer can only see the projection of P' on frame P. Moreover, the higher the relative speed between the two frames, the larger the angle of rotation and the smaller the projection of P' on P. Thus, the higher the relative speed, the smaller the values of measurements of events in P' made by the observer in P. However, the observer in frame P' would see no reduction in the values of measurements of events in P', no matter how fast P' is moving with respect to P (equivalently, no matter how large the angle between P' and P).3
The 3DSD clearly demonstrates why the coordinate axes of the moving frame P' are shown rotated toward each other in Minkowski's diagram: the observer in P, when obtaining lengths and time intervals in P', is actually measuring only the projection of those lengths and time intervals onto P. The two frames are truly independent spaces and the phrase "looking from the outside" can be taken literally. We observe here that the Lorentz transformation represents a real physical phenomenon, and is not just a rule for linking two observers' interpretations of one and the same space-time.
We are accustomed to thinking of rotations in space. But note that the rotation of P' away from P, while physical, does not depict a rotation of P' into some other spatial dimension, but rather into another kind of dimension. We are aware of this "rotation" of the space-time axes of the moving frame only through its speed relative to the stationary frame. Until now, this unexpected and unknown dimension into which P' rotates – let us call it u, for "unknown" – has been hidden from our view and therefore missing from our understanding not only of special relativity, but of reality itself!
Furthermore, because P' rotates through an angle that depends only on the relative speed of P' with respect to P (a parameter intrinsic to reference frames), this new dimension can be interpreted as a new degree of freedom for space-time frames themselves, rather than for objects therein. Needless to say, this rotation is invisible to us: We simply do not have sense organs to perceive dimension u.4 Therefore, we must take extreme care when trying to grasp the subtle distinction between a rotation of one space-time frame of reference (relative to another frame) into dimension u and a rotation of an object within a frame of reference. This important distinction may not be immediately obvious, and is therefore discussed at length in the upcoming book.
An important fact to retain is that: To be able to rotate freely into this extra dimension, the space-time of the moving frame must be different and separate from the space-time of the frame at rest. In other words, we cannot talk about just one space. There must be two physical frames for relative motion to occur, and not, as Einstein thought, merely two abstract mathematical coordinate systems. This suggests an important clue about the nature of reality: Reference frames are not merely mathematical coordinate systems. They are independent physical spaces in their own right!
We can conclude at this point that space-time frameworks -- interpreted as real and independent spaces -- are the fundamental elements of reality, and that their relationships determine the structure of the multispace. To truly appreciate this conclusion, however, the reader will have to also grasp the concept of dimensionless and timeless orthogonal spaces.
In fact, this concept is deceptively simple. On the last slide of the 3DSD slide show above, the relative speed of the moving frame is equal to the speed of light and plane P' is perpendicular (orthogonal) to plane P. The projection of plane P' on P is zero. Under these conditions, the Lorentz transformation tells us that all lengths and time intervals in the moving frame P', as measured by the observer in P, will be zero. In other words, according to the observer in P, one of the moving frame's spatial axes has collapsed to a point and all of its clocks have stopped.
The physical situation portrayed by the 3DSD is defined by the mathematical equations of special relativity. Thus any object in the moving reference frame will be seen from the outside as dimensionless (in the direction of relative motion) and eternal.
Minkowski's Legacy
Today, Hermann Minkowski is most famous for introducing a two-dimensional space-time diagram to explain the Lorentz transformation of Special Relativity. Over the last century, this intuitive visual aid has helped clarify Einstein's theory, significantly facilitating its dissemination to scientists and the general public. However, according to Minkowski, Einstein's former math teacher at the Polytechnic, Einstein clarified the physical significance of Lorentz's theory, but did not grasp the true meaning and full implication of the principle of relativity.5
Surprisingly, Minkowski also was the first to suggest that the world may be composed of an infinite number of spaces, but his writing was not explicit enough for other scientists to take his hint seriously.
Indeed, in 1908, Minkowski delivered his famous Cologne lecture "Raum und Zeit" (Space and Time). Explaining the natural laws of transformation between reference frames, he pointed out a subtle yet extremely important concept:
... We should then have in the world no longer "space," but an infinite number of spaces, analogously as there are in three-dimensional space an infinite number of planes.6
Sadly, Minkowski died soon after that lecture without having explained precisely what he meant by "an infinite number of spaces." Not understanding the extreme importance of Minkowski's declaration, no one – then or since – took his hint seriously, considering it nothing more than a "grandiose announcement" of the mathematician’s own relativity theory.
To learn how the 3D Space-Time Diagram changes the way we view reality, read the article "The Multispace Paradigm: Predictions About the True Structure of Reality."
ENDNOTES
1 You may want to open a second window to see the slideshow while you read this article.
3 Here we depart from the standard interpretation of relativity, in which each observer builds his own coordinate system and both sets of events and measurements refer to the same space-time. Instead, we adopt the notion of reference frame as set forth by Minkowski: Reference frames P and P' are actually independent physical spaces that overlap when the observers are at rest with respect to each other but diverge when they are not.
4 Remember that our bodies cannot directly experience "speed." To validate this assertion, recall that the Earth moves around the Sun without our feeling it.
5 Scott Walter, "Minkowski, Mathematicians, and the Mathematical Theory of Relativity,'' in H. Goenner, J. Renn, J. Ritter, T. Sauer (eds.), The Expanding Worlds of General Relativity (Einstein Studies, volume 7), pp. 45-86. Boston/Basel: Birkhäuser, 1999, page 61, or download it from here.
6 H. Minkowski, "Space and Time," in The Principle of Relativity, translated by W. Perrett and G.B. Jeffery (Dover Publications, Inc., 1952), p. 79.