1.3.2 Three dimensional spacetime model

The first step to deriving the distorted spacetime in three dimensions is to start from its representation in one dimension and try to develop it to the others two remaining. Figure 1.5a shows, the same rubber sheet model shown in Figure 1.3 but with only one dimension. As indicated, the figure suggests that at the middle, where the slope of spacetime is horizontal, there is an equilibrium point. Indeed, in real life, an equilibrium point exists at the centre of every mass, therefore, the drawing can be slightly improved if these two locations are overlapped as shown in Figure 1.5b.
In order to visualise spacetime as a three dimensional object, it is essential to immerse the mass into the rubber sheet, since all masses are actually immersed into spacetime. The first step, is then to insert the model into a coordinate system choosing the equilibrium point as the origin as shown in Figure 1.5c. Bearing in mind that the mass is buried into the spacetime medium, than the same thing seen in one direction would be also seen in all other directions. Starting with the horizontal, the same line above is mirrored below the axis, again shown in Figure 1.5c. Then, to better the idea of the mass being plunged into the rubber sheet, a number of layer, corresponding to distorted axis, can be added at both the top and the bottom of the origin, considering that the farther the layers are from the mass, the less the distortion caused by the mass would be, up to a point where the layer would be perfectly straight again (Figure 1.5d). Repeating the same procedure for the vertical direction, the resulting model is shown in Figure 1.5e, where, it is clearly visible, that the initial curvature of spacetime in the original model, has changed into a stretching inwards the mass. By repeating again the procedure, for the third axis, perpendicular to the page, the final three dimensions model is obtained as shown in Figure 1.6, where the mass would be at the centre inside the picture. Each line, represents a coordinate that gets distorted by the mass inside, by the same mysterious force that interacts with spacetime by curving it in the original rubber sheet model proposed by Einstein. The eight corners of the figure, are not distorted as they are further away from the mass than the sides which are instead distorted. This is because, being the mass spherical, then the distortion caused by it, all around it, must be spherical too. Moreover, the faces of the cube, should also be slightly distorted inward (concave). as their plane is not distant enough from the mass to not suffer any distortion.


 

(a)

(b)

(c)

(d)

(e)

Figure 1.5 - (a) One dimensional spacetime gravity model, (b) overlap of the two equilibrium points, (c) immerse the mass into the rubber sheet by first inserting the model into a coordinate system, (d) add layers of rubber and (e) repeat for vertical direction to obtain two dimensional model for gravity.

Figure 1.6  - Final three dimensional version of spacetime distortion around a spherical mass (left).

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