How Do Planck Scale Dimensions Define Universal Limits?

[Orion1]

Planck energy:
E_P = m_P c^2 = \sqrt{\frac{\hbar c^5}{G}}

Gravitational radius:
r_G = \frac{r_s}{2} = \frac{G m_P}{c^2}

Gravitational radius is equivalent to Compton wavelength:
r_G = \overline{\lambda}_C

\frac{G m_P}{c^2} = \frac{\hbar}{m_P c}

Planck force is a constant in the Einstein field equation:
F_P = \frac{E_P}{r_G} = m_P c^2 \left( \frac{c^2}{G m_P} \right) = \frac{c^4}{G} = \frac{8 \pi T_{\mu \nu}}{G_{\mu\nu}}

The maximum ratio of energy per gravitational length:
\boxed{\frac{c^4}{G} = \frac{8 \pi T_{\mu \nu}}{G_{\mu\nu}}}

Are Planck scale dimensions the maximum limits in the Universe?
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Reference:
http://en.wikipedia.org/wiki/Planck_force"
http://en.wikipedia.org/wiki/Planck_mass"
http://en.wikipedia.org/wiki/Planck_energy"

 

[Polyrhythmic]

Where did you get that expression for the quotient of the Stress-enery-tensor and the Einstein-tensor from? As far as I know, it's not valid to divide two tensor-valued quantities.

 

[Orion1]

Where did you get that expression for the quotient of the Stress-enery-tensor and the Einstein-tensor from?

From Einstein's field equation:
G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}

As far as I know, it's not valid to divide two tensor-valued quantities.

It is not possible to divide tensors. However, It is possible to divide the solutions after the tensors have been solved for a specific solution.

For example:
\boxed{\frac{c^4}{G} = \frac{8 \pi T_{r r}}{G_{r r}}} \; \; \; \mu = \nu

Are Planck scale dimensions the maximum limits in the Universe?
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