3 Null Geodesic Motion in n Dimensions

Photons propagate at a spatial speed $c$ in every frame of reference which means that their time-speed $\chi$ is always zero. This effectively eliminates one dimension from the geometric environment of the photon.

From our dimensional viewpoint in $X_4$ we thus may consider the photon to have its properties position and speed defined in a 3D subspace $X_3$ (note that the photon is treated here as a particle and not as a (4D) electromagnetic wave). The photon follows null geodesics in $X_4$ that are equivalent to timelike geodesics from the dimensional viewpoint of $X_3$ (the viewpoint of the photon itself). The time-speed of the photon may therefore be expressed in $X_3$ as

$$\chi_3=dx_3/dx_4$$ (7)

conform the generalized definition given in Eq. (5).

The view from the rest frame of the photon in $X_3$ (which is impossible in $X_4$ but possible in $X_3$ ) closely resembles the view that we have of $X_4$ . The photon's own spatial speed $c$ , as measured in $X_4$ , is seen as the flow of time in $X_3$ . In $X_3$ photons can have spatial speeds relative to each other ranging from zero to $c$ . The main difference is that in $X_3$ relative spatial motion between photons is only measurable in two dimensions, $x_1$ and $x_2$ . This conforms to the generalized definition for spatial speeds in $X_3$ according to Eq. (6):

$$v_a=dx_a/dx_4 \;\;(a=1,2)$$ (8)

Similarly, the timelike geodesics for mass particles in $X_4$ can be regarded null geodesics from the perspective of $X_5$ (see also [4] for a similar treatment of geodesics in Kaluza-Klein like theories). This apparent hierarchy between geodesic motion for mass particles and photons (an observation that is also made in [5]) suggests a similar hierarchy between the fields for electromagnetism and gravity. The electromagnetic field should then be describable in terms of a curved Riemannian manifold with one less dimension as gravity.

The expressions for speeds of mass particles in $X_4$ , $\chi_4=cdx_4/dx_5$ and $v_i=dx_i/dx_5$ , show that $x_5$ must be an essential part of the field that propagates mass particles, i.e., gravity must be 5D. This is consistent with the expression $\chi_3=dx_3/dx_4$ where $x_4$ is part of the electromagnetic field that propagates photons. Gravity being 5D thus allows the logic association of electromagnetism with four dimensions, which shows from the four-vector for potential $A^{\mu}=\left(\phi,c \mathbf{A}\right)$ (see [2] that shows the Euclidean nature of this 4-vector, as opposed to the Minkowski nature of other 4-vectors) and the 4D tensor notation for E and B.

Extending general relativity to five dimensions is not uncommon in Kaluza-Klein like theories (using the Minkowski metric). The universal speed $c$ for objects in 4D, as found in the Euclidean treatment of special relativity in [1], also holds in general relativity based on five Euclidean dimensions if it is assumed that, due to the curved space-time near a massive object, an additional rotation of the universal speed $c$ towards the axis of the fifth dimension takes place from the perspective of an observer at infinity who uses a flat 5D coordinate system. Individual velocity components for a particle falling radially towards a massive object would then be the coordinate speed in 3-dimensional space,

$$v(r)=\{1-2MG/(rc^2)\}\sqrt{2MG/r}$$ (9)

the speed in the proper time dimension $x_4$ ,

$$\chi(r)=c\sqrt{1-2GM/(rc^2)}$$ (10)

and the speed in $x_5$ ,

$$v_5(r)=\sqrt{c^2-\chi(r)^2-v(r)^2}$$ (11)

Figures 1 and 2 illustrate the developments of these velocity components for a radially falling object.

Figure 1: Graph of velocity components in space: $v(r)$ , proper time: $\chi (r)$ , and $x_5$ : $v_5(r)$ for an object falling radially towards a black hole.

The dotted vertical line intersecting the graphs in Fig. 1 corresponds to the position of the velocity vector of magnitude $c$ in Fig. 2.

Figure 2: Velocity of magnitude $c$ in 5 dimensions for an object falling radially towards a black hole.

At the Schwarzschild radius the rotation will have completed, resulting in the predicted zero speed in both space and time, while in all geodesics the 5D velocity vector maintains magnitude $c$ . This process may be identical (translated to 4D) to the way photons approach electrical charges. In that case the photon's velocity vector rotates towards the fourth dimension when nearing an electrical charge. Its spatial speed reduces to zero at the moment that it is absorbed by the charge and the velocity vector is fully rotated into the fourth dimension. Mass particles falling into black holes is then equivalent to photons being absorbed by electrical charges.