Four-dimensional space

...

when I know that even flat spacetime is 4D.

[laughs]

And I've seen the pictures.

Throw away two dimensions just for now, so you have one dimension of space and one of time.

Now you have a surface.

So we're saying spacetime is just 2D dot 4D, and then gravity causes that surface to bend, to warp.

But it's a, it's a major approximation that, that it's, it's, it...

Here the contraction of moving bodies follows from the two fundamental principles of the theory without the introduction of particular hypotheses.

and as the prime factor involved in this contraction we find not the motion in itself to which we cannot attach any meaning but the motion with respect to the body of reference chosen in the particular case and point thus for a coordinate system moving with the earth the mirror system of michelson and morley is not shortened but it is shortened for a coordinate system which is at rest relatively to the sun section Section 17 Minkowski's Four-Dimensional Space the non-mathematician is seized by mysterious shuddering when he hears of four-dimensional things by feeling not unlike that awakened by thoughts of the occult and yet there is no more commonplace statement than that the world in which we live is a four-dimensional space-time continuum space is a three-dimensional continuum by this we mean that it is possible to describe the position of a point at rest by means of three numbers or coordinates x y z and that there is an indefinite number of points in the neighborhood of this one the position of which can be described by coordinates such as x sub one y sub one z sub one which may be as near as we choose the respective values of the coordinates x y z of the first point in virtue of the latter property we speak of a continuum and owing to the fact that there are three coordinates we speak of it as being three-dimensional Similarly, the world of physical phenomena, which was briefly called world by Minkowski, is naturally four-dimensional in the space-time sense, for it is composed of individual events, each of which is described by four numbers, namely three space coordinates x, y, z, and a time coordinate, the time value t.

The world is, in this sense, also a continuum, for to every event there are as many neighboring events, realized or at least thinkable as we care to choose, the coordinates x sub 1, y sub 1, z sub 1, t sub 1, of which differ by an indefinitely small amount from those of the events x, y, z, t originally considered.

that we have not been accustomed to regard the world in this sense as a four-dimensional continuum is due to the fact that in physics before the advent of the theory of relativity time played a different and more independent role as compared with the space coordinates it is for this reason that we have been in the habit of treating time as an independent continuum as a matter of fact according to classical mechanics time is absolute i e it is independent of the position and the condition of the motion of the system of coordinates We see this expressed in the last equation of the Galilean transformation, t prime equals t.

Pure space distance of two events with respect to k results in time distance of the same events with respect to k'.

But the discovery of Minkowski... which was of importance in the formal development of the theory of relativity, does not lie here.

It is to be found rather in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity and its most essential formal properties shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space.

Begin footnote.

I realize that as a psychiatrist you are interested in minds, in living beings, rather than in dimensional planes.

But I fear you will find no minds to study in the fourth dimension.

There aren't any there.

Professor Galt paused, peered from beneath bushy white brows out over the laboratory.

Galt hoped sourly that the young man was actually working, and not just drawing more of his absurd senseless designs amidst the mathematical computations.

Your proof, Dr. Pillbot broke into his thoughts insistently, is purely negative, Professor.

How can you know there are no beings in the fourth dimension unless you actually enter this realm to see for yourself? Professor Galt stared at the fat, puffy face of his visitor and snorted loudly.

I am afraid, Pillbot, you do not comprehend the impossibility of such a passage.

Now, it is very remarkable that this is so extensively overlooked, continued the time traveller, with a slight accession of cheerfulness.

Really this is what is meant by the fourth dimension, though some people who talk about the fourth dimension do not know they mean it.

It is only another way of looking at time.

There is no difference between time and any of the three dimensions of space except that our consciousness moves along it.