LOG#206. Multitemporal theories.

Newton’s gravity reads:

    \[F_N=G_N\dfrac{Mm}{R^2}\]

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

    \[F_{D}=G_D\dfrac{Mm}{R^{D-2}}=G_D\dfrac{Mm}{R^{d-1}}\]

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

    \[F_N=F_D\]

    \[\boxed{G_N=\dfrac{G_D}{R^n}}\]

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):

    \[\boxed{M_P^2(4D)=M_D^2R^n}\]

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

    \[\boxed{G_N=\dfrac{G_D}{V_n}}\]

    \[\boxed{M^2_D V_n=M^2_P}\]

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

    \[\boxed{F(XT)=G^{xt}_{(ij)}\dfrac{M^iM^j}{R^d}\cos^2\theta}\]

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

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

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

so

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

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

    \[\boxed{G_{4D,eff}=G_N\cos^2\theta}\]

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

    \[\boxed{M_{D,eff}^2=M_{4D}^2\cos^2\theta}\]

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

    \[\boxed{G_{4D,eff}^{(ij)}=G_N^{(ij)}\cos^2\theta}\]

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

Questions:

  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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