LOG#206. Multitemporal theories.

Newton’s gravity reads:


In extra dimensions, D=d+1, d-1=2+n, it reads


For extra dimensions, if their size are much smaller than considered distances, R>>R_{XD}, then by matching



So, gravitational is weak in our scale because it gets diluted through extra dimensions. Real Planck scale gravity is much stronger. In terms of mass scales (large ADD scenario):


More generally, we can substitute R^n by a volumen V_n:


    \[\boxed{M^2_D V_n=M^2_P}\]

What if you get extra time-like dimensions. Let N=n+1+d the number of dimensions. Then,


The proof is also straightforward:

    \[F^{xt}=G_D\dfrac{\varepsilon_i M^{ij}\varepsilon_j}{R^d}\]

with \varepsilon_i the time vectors, such that

    \[\cos\theta=\varepsilon_i\cdot \varepsilon_j/\vert\varepsilon_i\vert\vert\varepsilon_j\vert\]

    \[F^{xt}=G^{(ij)}\dfrac{M^i_{\; k}M^{\; k j}}{R^d}\]


    \[\boxed{F^{xt}_{ij}=G_{(ij)}\dfrac{M^i_{\; k}M^{\; k j}}{R^d}\cos^2\theta}\]

where, with d=2+s, N=n+1+d=n+1+2+s=n+2+s. Therefore,


and gravity is “small” due to the almost orthogonality of time vectors. Equivalently,


We can indeed combine the extradimensional space-like case with the time-like case:


    \[\boxed{M_P^{2 (ij)}=M_D^{2 (ij)} V^s \cos^2\theta}\]


  1. What are more interesting, extra time-like or extra space-like dimensions?
  2. Why extra time-like dimensions are IMPORTANT despite being generally neglected by theorists, excepting a few excepcional cases?
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