Newton’s gravity reads:
![\[F_N=G_N\dfrac{Mm}{R^2}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-f18e28fb357403e62e8413fc5e118d8a_l3.png "Rendered by QuickLaTeX.com")
In extra dimensions, 
, 
, it reads
![\[F_{D}=G_D\dfrac{Mm}{R^{D-2}}=G_D\dfrac{Mm}{R^{d-1}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-9055a9c9dd1306224b254a10ac352a64_l3.png "Rendered by QuickLaTeX.com")
For extra dimensions, if their size are much smaller than considered distances, 
, then by matching
![\[F_N=F_D\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-155740d56dbd350dc4886fe2ad296412_l3.png "Rendered by QuickLaTeX.com")
![\[\boxed{G_N=\dfrac{G_D}{R^n}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-d72864e8a16dc97c7a2feb6955cd514b_l3.png "Rendered by QuickLaTeX.com")
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):
![\[\boxed{M_P^2(4D)=M_D^2R^n}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-8894875c162788c87686a9234be97d4f_l3.png "Rendered by QuickLaTeX.com")
More generally, we can substitute 
by a volumen 
:
![\[\boxed{G_N=\dfrac{G_D}{V_n}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-9462d3dc0612211ab2d0ef7d380d68ee_l3.png "Rendered by QuickLaTeX.com")
![\[\boxed{M^2_D V_n=M^2_P}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-949480ec3a608a44ed1a1800e2fe3993_l3.png "Rendered by QuickLaTeX.com")
What if you get extra time-like dimensions. Let 
the number of dimensions. Then,
![\[\boxed{F(XT)=G^{xt}_{(ij)}\dfrac{M^iM^j}{R^d}\cos^2\theta}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-bc65699a3b39516a8c89dfa69087f2d7_l3.png "Rendered by QuickLaTeX.com")
The proof is also straightforward:
![\[F^{xt}=G_D\dfrac{\varepsilon_i M^{ij}\varepsilon_j}{R^d}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-4a721d7eac2cb1c121c707e3c486b083_l3.png "Rendered by QuickLaTeX.com")
with 
the time vectors, such that
![\[\cos\theta=\varepsilon_i\cdot \varepsilon_j/\vert\varepsilon_i\vert\vert\varepsilon_j\vert\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-dfe2ba2a15fab5e8e5b7420aa81d64e6_l3.png "Rendered by QuickLaTeX.com")
![\[F^{xt}=G^{(ij)}\dfrac{M^i_{\; k}M^{\; k j}}{R^d}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-d906d799307b18d2cacf91795729c7ac_l3.png "Rendered by QuickLaTeX.com")
so
![\[\boxed{F^{xt}_{ij}=G_{(ij)}\dfrac{M^i_{\; k}M^{\; k j}}{R^d}\cos^2\theta}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-4b645c545913a6a3a7df7f7bd51c064c_l3.png "Rendered by QuickLaTeX.com")
where, with 
, 
. Therefore,
![\[\boxed{G_{4D,eff}=G_N\cos^2\theta}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-86f4f21626d1540439f8858940b4c9f3_l3.png "Rendered by QuickLaTeX.com")
and gravity is “small” due to the almost orthogonality of time vectors. Equivalently,
![\[\boxed{M_{D,eff}^2=M_{4D}^2\cos^2\theta}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-5d4a14a89cfe500c36752b606a105b0b_l3.png "Rendered by QuickLaTeX.com")
We can indeed combine the extradimensional space-like case with the time-like case:
![\[\boxed{G_{4D,eff}^{(ij)}=G_N^{(ij)}\cos^2\theta}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-c3e557f19de68b086ca6a6d305a194a8_l3.png "Rendered by QuickLaTeX.com")
![\[\boxed{M_P^{2 (ij)}=M_D^{2 (ij)} V^s \cos^2\theta}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-db56428a0efb74fbf7196ee767e27b59_l3.png "Rendered by QuickLaTeX.com")
Questions:
- What are more interesting, extra time-like or extra space-like dimensions?
- Why extra time-like dimensions are IMPORTANT despite being generally neglected by theorists, excepting a few excepcional cases?
