simple explanation for Time Dilation & SWE ?

[Widdekind]

First, the Relativistic energy equation:

 

E

2

= p

2

+ m

2

 

treats rest-mass like a "hyper-momentum", in an extra spatial dimension:

 

E

2

= p

xyz

2

+ p

w

2

 

That extra "w" dimension can be construed, as the hyper-spatial "thickness" of the fabric of space-time. The fabric of space-time may have an "inside surface", and an "outside surface". And, the wave-functions of quanta may reside in between both said surfaces, like ice cream between the wafers of an ice cream sandwich. In reduced-dimensional visualization, in (1+1)D, the space-time fabric of our universe may resemble a "vase". In this hypothesis, that "vase" would have some "hyper-thickness", and would not be an infinitely thin membrane. Wave-functions of quanta hypothetically "slosh back and forth", reflecting from the "inside" & "outside" surfaces of the space-time fabric, which hypothetically acts as a wave-guide:

 

 

Mass-less photons propagate, at the speed of light ©, entirely through time & space. Massive particles, at rest, propagate, at light-speed, entirely through time and the hyper-spatial "thickness" of the fabric of space-time. As massive particles are accelerated, their "absolute" hyper-spatial velocity rotates, from entirely "across" the fabric of space-time, bouncing back and forth in the "w" dimension; to entirely through the fabric of space-time, in the "xyz" dimensions. When accelerated to spatial velocity (v), the particle zig-zags across and through the space-time fabric, at some angle to the "w" axis:

 

[math]v_{xyz} = c \; sin(\theta)[/math]

 

[math]v_w = c \; cos(\theta)[/math]

 

Because some of the particle's velocity is "used up" propagating through space, the particle propagates "across" space (in the "w" thickness dimension) more slowly than when at rest. The longer "bounce time" explains (Special) Relativistic time-dilation. If the "thickness" of space-time is [math]\delta w[/math]:

 

[math]\delta t_{bounce} = \frac{\delta w}{ c \; cos(\theta) } = \frac{\delta t_0}{\sqrt{1 - \beta^2}} = \gamma \; \delta t_0[/math]

 

But, why would slower "sloshing side-to-side across space-time" make clocks run slower, i.e. make quantum wave-functions evolve more slowly? This picture implies, that evolution of wave-functions somehow requires "bounces" off of the bounding surfaces of the space-time fabric. According to the SWE, the evolution of wave-functions is proportional to their energies:

 

[math]\delta \Psi = \frac{\delta t}{\imath \hbar} \hat{H} \Psi[/math]

 

And, in (General) Relativity, energies cause curvatures into the fabric of space-time. So, perhaps the energies of quanta induces "wrinkles" in the (inner & outer) "skins" of space-time; and when their wave-functions "slosh" up against said wrinkled surfaces, they reflect with distortions & diffractions, that cause wave-functions to evolve, spread out, etc. Zooming into the (1+1)D space-time fabric visualized above, looking at a short segment of space, at a single slice of time, whilst emphasizing said segment's hyperspatial thickness; the presence of a particle with energy may "wrinkle" the bounding "skin" surfaces of the space-time fabric:

 

[math]= \; \longrightarrow \; \approx[/math]

 

Then, as the particle's wave-function bounces back and forth, across the hyper-spatial thickness of the space-time fabric [math]\left( \uparrow \downarrow \right)[/math] the "wavy" space-time fabric induces distorting diffractions into the wave-function, after every bounce & reflection. Thus, the rate of reflections & bounces [math]\left( \delta t = \gamma \; \delta t_0 \right)[/math] determines the rate at which the wave-function evolves. Wave-functions evolve by one unit of change per bounce:

 

[math]\delta \Psi = \frac{\delta t}{\imath \hbar} \hat{H} \Psi[/math]

 

[math]\delta \Psi = \frac{\gamma \; \delta t_0}{\imath \hbar} \hat{H} \Psi[/math]

 

Wave-functions evolve slower, at speed, because they bounce back-and-forth across the space-time fabric more slowly; and their infrequent sloshings side-to-side afford less rapid evolution. One must wait longer [math]\left( \delta t_0 \longrightarrow \gamma \; \delta t_0 \right)[/math] to observe the same change [math]\left( \delta \Psi \right)[/math].

 

This same picture can account, too, for gravitational time-dilation. For, around a massive object, space "sags" according to the Flamm paraboloid; the hyper-spatial height (w) of the space-time fabric is:

 

[math]w® = 2 R_S \sqrt{\frac{r}{R_S} - 1}[/math]

 

per the rubber sheet analogy. If you imagine, that wave-functions always bounce "up and down", whether that rubber sheet lies flat horizontally; or sags down somewhat vertically; then for a given transverse thickness [math]\delta w[/math] of the rubber sheet, the vertical distance the wave-function must actually propagate in between bounces is:

 

[math]\delta l \; cos(\theta) = \delta w[/math]

(angle between fixed "vertical" and normal to space-time fabric)

 

[math]tan(\theta) = \frac{dw}{dr}[/math]

(angle between "horizontal" and tangent to the space-time fabric, equal to the above by geometry)

 

[math]\delta t = \frac{\delta l}{c} = \frac{\delta w}{c \; cos(\theta)} = \frac{\delta w}{c} sec(\theta)[/math]

 

[math] = \delta t_0 \sqrt{1 + tan^2(\theta)}[/math]

 

[math] = \delta t_0 \frac{1}{\sqrt{1 - \frac{R_S}{r} }}[/math]

 

which is the correct formula, from the Schwarzschild metric. Thus, gravity can be construed as simulating speed, since curved space-time also creates an angle between the hyper-spatial / spatial propagation of wave-functions; and the wave-guide-like, skin-like surfaces, of the fabric of space-time. Speed rotates the former w.r.t. the latter; whereas gravity curves the latter w.r.t. the former. Either way, wave-functions bounce back-and-forth less frequently; and since said bounces are what induce distorting diffractions into the wave-functions, so slower bouncing implies slower wave-function evolution, i.e. the appearance of "time-dilation".

 

This simple picture, of space-time being a fabric with non-zero hyper-spatial "thickness" [math]\left( \delta w \right)[/math], explains both the SWE; and Special & General Relativistic time dilation.