- #1
Brute Force
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Could somebody explain me the following:
According to GR time dilation due to gravitational field is expressed as:
T[itex]_{g}[/itex]=T[itex]_{f}[/itex]*[itex]\sqrt{1-\frac{2GM}{rc^{2}}}[/itex]
where Tg is time with gravitation,
Tf is time somewhere without gravitation
G - gravitational constant
M - mass
r - radial coordinate of observer
and c - light speed.
Lets assume two explorers are working on two different planets with the same gravitation.
My understanding is that gravitation could be expressed as g=G[itex]\frac{M}{r^{2}}[/itex].
Mass M for spherical planet is equals to: M=[itex]\frac{4}{3}[/itex][itex]\pi[/itex]r[itex]^{3}[/itex][itex]\rho[/itex], where [itex]\rho[/itex] is density.
If two gravitations are the same then:
g[itex]_{1}[/itex]=g[itex]_{2}[/itex] and [itex]\rho[/itex][itex]_{1}[/itex]r[itex]_{1}[/itex]=[itex]\rho[/itex][itex]_{2}[/itex]r[itex]_{2}[/itex] or [itex]\rho[/itex]r=const
Getting back to the time dilation formula and replacing M with V*[itex]\rho[/itex]:
T[itex]_{g}[/itex]=T[itex]_{f}[/itex][itex]\sqrt{1-\frac{2G}{c^{2}}\frac{4}{3}\pi{r^{2}\rho}}[/itex]
After replacing all constants with k (remember that [itex]\rho[/itex]r is also a constant:
T[itex]_{g}[/itex]=T[itex]_{f}[/itex][itex]\sqrt{1-k*r}[/itex]
What a surprise: same gravitation, but different time dilation! (assuming planets radii are different)
Did I missed something?
Thanks.
According to GR time dilation due to gravitational field is expressed as:
T[itex]_{g}[/itex]=T[itex]_{f}[/itex]*[itex]\sqrt{1-\frac{2GM}{rc^{2}}}[/itex]
where Tg is time with gravitation,
Tf is time somewhere without gravitation
G - gravitational constant
M - mass
r - radial coordinate of observer
and c - light speed.
Lets assume two explorers are working on two different planets with the same gravitation.
My understanding is that gravitation could be expressed as g=G[itex]\frac{M}{r^{2}}[/itex].
Mass M for spherical planet is equals to: M=[itex]\frac{4}{3}[/itex][itex]\pi[/itex]r[itex]^{3}[/itex][itex]\rho[/itex], where [itex]\rho[/itex] is density.
If two gravitations are the same then:
g[itex]_{1}[/itex]=g[itex]_{2}[/itex] and [itex]\rho[/itex][itex]_{1}[/itex]r[itex]_{1}[/itex]=[itex]\rho[/itex][itex]_{2}[/itex]r[itex]_{2}[/itex] or [itex]\rho[/itex]r=const
Getting back to the time dilation formula and replacing M with V*[itex]\rho[/itex]:
T[itex]_{g}[/itex]=T[itex]_{f}[/itex][itex]\sqrt{1-\frac{2G}{c^{2}}\frac{4}{3}\pi{r^{2}\rho}}[/itex]
After replacing all constants with k (remember that [itex]\rho[/itex]r is also a constant:
T[itex]_{g}[/itex]=T[itex]_{f}[/itex][itex]\sqrt{1-k*r}[/itex]
What a surprise: same gravitation, but different time dilation! (assuming planets radii are different)
Did I missed something?
Thanks.