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## Main Question or Discussion Point

Consider a mode of vacuum zero-point energy at a point in space. Its energy [itex]E[/itex] is related to its frequency [itex]f[/itex] by

[tex] E = \frac{1}{2}h f.[/tex]

In terms of the mode oscillation period [itex]\Delta t[/itex] the energy is given by

[tex] E = \frac{1}{2}\frac{h}{\Delta t}. [/tex]

Now lets us imagine that [itex]\Delta t[/itex] becomes smaller and smaller. Therefore the mass/energy of the vacuum mode will become larger and larger. Eventually that point in space will collapse in on itself and become a blackhole.

The size of a blackhole is its Schwarzschild radius [itex]\Delta x[/itex] which is given by

[tex]\Delta x = \frac{2GM}{c^2}[/tex]

By using the relation [itex]M=E/c^2[/itex] we can combine the above expressions to find that

[tex]\Delta x \Delta t = \frac{G h}{c^4}[/tex]

Does this relationship indicate that spacetime is quantized?

I presume this expression is Lorentz invariant because when one transforms to another inertial frame the dilation of the time interval is cancelled by the contraction of the length interval.

In a FRW universe the length interval is proportional to the scale factor. In order that the product of the time and space intervals be constant this seems to imply that the time interval in a FRW universe should be

[tex] E = \frac{1}{2}h f.[/tex]

In terms of the mode oscillation period [itex]\Delta t[/itex] the energy is given by

[tex] E = \frac{1}{2}\frac{h}{\Delta t}. [/tex]

Now lets us imagine that [itex]\Delta t[/itex] becomes smaller and smaller. Therefore the mass/energy of the vacuum mode will become larger and larger. Eventually that point in space will collapse in on itself and become a blackhole.

The size of a blackhole is its Schwarzschild radius [itex]\Delta x[/itex] which is given by

[tex]\Delta x = \frac{2GM}{c^2}[/tex]

By using the relation [itex]M=E/c^2[/itex] we can combine the above expressions to find that

[tex]\Delta x \Delta t = \frac{G h}{c^4}[/tex]

Does this relationship indicate that spacetime is quantized?

I presume this expression is Lorentz invariant because when one transforms to another inertial frame the dilation of the time interval is cancelled by the contraction of the length interval.

In a FRW universe the length interval is proportional to the scale factor. In order that the product of the time and space intervals be constant this seems to imply that the time interval in a FRW universe should be

*inversely proportional*to the scale factor.