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## Main Question or Discussion Point

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.