Need help understanding an equation

[assafwei]

Hi,

Im reading an article (The meaning of einstein's equation : http://math.ucr.edu/home/baez/einstein/node6a.html" [Broken] )

and cant understand the development of one of the equations, the equation is attached as an image, I cant understand the transition from the left side of the equation to the middle.
Can anyone help?

Thanks.

 

[George Jones]

Taylor's theorem.

$$\delta V = V \left( t + \delta t \right) - V \left( t \right) = \dot{V} \left( t \right) \delta t + \frac{1}{2!} \ddot{V} \left( t \right) \left (\delta t \right)^2 + \dots$$

Neglect higher order terms, take $t = 0$, and use $V \left( 0 \right) = 0$.

 

[HallsofIvy]

In other words, that is expanding V(x) in a Taylor's series about x= t, then replacing x by $t+ \delta t$.

Actually, I don't believe they are taking "V(0)= 0". Since they are taking the Taylor's series about x= t, the constant term would be V(t) and they are subtracting off V(t).

[tex]V(x)= V(t)+ \dot{V}(t)(x- t)+ \frac{\ddot{V}(x- t)^2+ \cdot\cdot\cdot[/itex]
$$V(t+ \delta t)= V(t)+ \dot{V}(t)(t+ \delta t- t)+ \frac{\ddot{V}(t)}{2}(t+ \delta t- t)^2+ \cdot\cdot\cdot$$
$$V(t+\delta t)= V(t)+ \dot{V}(t)\delta t+ \frac{\ddot{V}(t)}{2}(\delta t)^2+ \cdot\cdot\cdot$$
so
$$V(t+ \delta t)- V(t)= \dot{V}(t)\delta t+ \frac{\ddot{V}(t)}{2}(\delta t)^2+ \cdot\cdot\cdot$$

 

[George Jones]

Actually, I don't believe they are taking "V(0)= 0".

Oops, I meant to write

$$\dot{V} \left( 0 \right) = 0.$$

From the link given in the original post:

"Fill the interior of the sphere with test particles, all of which are initially at rest relative to the planet. ... Thus, after a short time $\delta t$ has elapsed, ..."

 

[HallsofIvy]

Oh, I see. I hadn't looked at the original attachment.