does multiplying E = MC^2 and time dilation have any significance?
I have several questions: 1. who came up with this and is there any analysis of this math and physics?
1. Does this relationship have any realizations or conclusions that can be drawn?
here are some links for those interested
Time Dilation Link
Mass Energy Link
Here is the math:
$$(1)\qquad E = m{c}^2$$ $$(2)\qquad \Delta t_1 = \frac{\Delta t_2}{\sqrt{1-\left({\left(\frac{v}{c}\right)}^2\right)}}$$
for easily seeing the binomial expansion of $(2)$
$$let \quad x = {-\left({\left(\frac{v}{c}\right)}^2\right)}$$
rearranging of $(2)$ $$\Delta t_1 = \frac{\Delta t_2}{\sqrt{1+x}}$$
$$\frac{\Delta t_1}{\Delta t_2} = \frac{1}{\sqrt{1+x}}$$
$$\frac{\Delta t_1}{\Delta t_2} = \left({1+x}\right)^{-1 \over 2}$$
$$(3) \qquad \left({1+x}\right)^{-1 \over 2}$$
Applying the binomial expansion on equation $(3)$ to get the following: Fractional power binomial expansion link
$$\left({1+x}\right)^{-1 \over 2} = 1{x}^0 + -\frac{1}{2}{x}^1 + \frac{3}{8}{x}^2 + -\frac{5}{16}{x}^3 + \frac{35}{128}{x}^4+ \cdots$$
essentially the coefficients are $${n \choose q}\quad or \quad{-\frac{1}{2} \choose q}$$ where n is $-\frac{1}{2}$ and $q$ incriments up by 1 starting at 0, I think.
also
$$ -1 \lt x \lt 1$$
now substituting $$x = {-\left({\left(\frac{v}{c}\right)}^2\right)}$$
$$\left({1-\left({\left(\frac{v}{c}\right)}^2\right)}\right)^{-1 \over 2} = 1\bullet{\left({-\left(\frac{v}{c}\right)}^2\right)}^0 + -\frac{1}{2}\bullet{\left({-\left(\frac{v}{c}\right)}^2\right)}^1 + \frac{3}{8}\bullet{\left({-\left(\frac{v}{c}\right)}^2\right)}^2 + -\frac{5}{16}\bullet{\left({-\left(\frac{v}{c}\right)}^2\right)}^3 + \frac{35}{128}\bullet{\left({-\left(\frac{v}{c}\right)}^2\right)}^4+ \cdots$$
"$\bullet$" symbol here is just a multiply,because the parenthesis are getting crowded here. 565