Time dilation due to gravity (GR)-

[tex]d\tau=dt\sqrt{1-\frac{2M}{r}}[/tex]

where [itex]M=Gm/c^2[/itex]

Time dilation due to velocity (SR)-

[tex]d\tau=dt\sqrt{1-\frac{v^2}{c^2}}[/tex]

Quantities-

Earth's mass- 5.9736e+24 kg

Earth's (mean) radius- 6.371e+6 m

Satellite's altitude- 2e+7 m

Satellite's velocity- 3.889e+3 m/s
...

Continued from Critical Density

Equations that demonstrate a flat and accelerating universe-

Friedmann equation-

[tex]H^2=\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3}[/tex]

Friedmann acceleration equation-

[tex]\dot{H}+H^2=\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3P}{c^2}\right)+\frac{ \Lambda c^2}{3}[/tex]

where [itex]P[/itex] is pressure and the...

Einstein field equations (EFE)-

[tex]G_{\mu\nu}+g_{\mu\nu}\Lambda=\frac{8\pi G}{c^4} T_{\mu\nu}[/tex]

G_μν is the Einstein tensor of curvature (spacetime), g_μν is the metric tensor, Λ is the cosmological constant, [itex]8\pi[/itex] is the concentration factor and T_μν is the energy tensor of matter (matter energy)

c and G are introduced to convert the quantity (which is expressed in physical units) to geometric units (G/c⁴ is used to convert units of...

Note: The following only applies at the equatorial plane.

local gravitational acceleration for an object in orbit at the equatorial plane-

[tex]
\begin{flalign}
&a(U)=-\gamma^2[g-2v\theta_{\hat{\phi}}-v^2k_{(\text{lie})}]\\[6mm]

&\vec{g}=-\vec{a}(n)\\[6mm]

&a(n)_{\hat{r}}=\frac{M\left[(r^2+a^2)^2-4a^2Mr\right]}{\sqrt{\Delta}\,r^2(r^3+a^2r+2Ma^2)}\\[6mm]

&\theta_{\hat{\phi}\,\hat{r}}=\frac{Ma(3r^2+a^2)}{r...

In mathematical physics, a closed timelike curve (CTC) is a worldline in a Lorentzian manifold, of a material particle in spacetime that is "closed," returning to its starting point.

'Inside the inner horizon (of a charged/rotating black hole) there is a toroidal region around the ring singularity within which the light cone in t-φ coordinates opens up to the point that φ as well as t are time-like coordinates. The direction of increasing proper time along t is...