Time, Time Speed, Aging

If we look at the expression for gravity

\(g=-\cfrac { 1 }{ 2 } .\cfrac { d{ v }_{ t }^{ 2 } }{ dx }\)

Integrating both

\(-2\int { { g } } dx=\int { d{ v }_{ t1 }^{ 2 } } ={ v }_{ t1 }^{ 2 }\)

The left hand side is negative given that g decreases with increasing x.

We have seen that g is of the form, \(g=-{g}_{o}{e}^{-\cfrac{{g}_{o}{r}_{e}}{{G}_{o}}(x)}\), therefore

\(\int { { g } } dx=\cfrac{{G}_{o}}{{r}_{e}}{e}^{-\cfrac{{g}_{o}{r}_{e}}{{G}_{o}}(x)}+A\)

And,

\({ v }_{ t1 }^{ 2 }=C-{\cfrac{2{G}_{o}}{{r}_{e}}}{e}^{-\cfrac{{g}_{o}{r}_{e}}{{G}_{o}}(x)}\)

Since, \({ v }_{ t1 }^{ 2 }={c}^{2}\) when \(x\rightarrow\infty\), where \({d}_{s} \)is at normal space density. \(C={c}^{2}\) and so,

\({v}_{t}=\sqrt{{c}^{2}-{\cfrac{2{G}_{o}}{{r}_{e}}}{e}^{-\cfrac{{g}_{o}{r}_{e}}{{G}_{o}}(x)}}\)

We have an expression similar to before except \(\cfrac{1}{(x+{r}{e})^2}\) is replaced with \({e}^{-\cfrac{{g}_{o}{r}_{e}}{{G}_{o}}(x)}\).

Similarly, time dilation \(\gamma\),

\(\cfrac{{v}_{t}}{c}=\gamma=\sqrt{1-{\cfrac{2{G}_{o}}{{c}^{2}{r}_{e}}}{e}^{-\cfrac{{g}_{o}{r}_{e}}{{G}_{o}}(x)}}\)

 In a gravity field time speed is less than time speed at normal space density, \(c\). We experience a shorter second and so, age faster than when gravity is zero where time speed is higher. At higher time speed very second is longer than at slower time speed, if 10 seconds at lower time speed fit into 9 seconds in higher time speed then those at slower time speed is aging 10% faster. Most will think the opposite, that higher time speed mean "time flies" and so age faster or that a second is used up faster. Higher time speed produces a longer second based on a standard second at some standard time speed. A lower time speed travels a shorter second. In the same standard time frame, there are more "short time speed" seconds than "high speed time" seconds.  Those at high time speed age less.

Imagine Time has markers make out in space, as space stretches out the time markers stretches out too.  X per second means X marks one second.  It is confusing because X is also measured in seconds.  Time speed should be measured based on some standard time interval for a second, and so X has to be adjusted by a factor inversely proportional its time speed, when taking physics measurements across different time speeds. Do not be confused by slow-motion cinematic effects, at different time speeds, all will experience time in the same way, within that time speed.

All physics have common expressions across all time speeds within that time speed.  The factor, \({\cfrac{{c}}{{v}_{t}}}\) adjustment is needed only when one has to define a standard time interval at speed c, in order to discuss across different time speeds.