Gravitational Time Dilation

Terms: Gravitational Time Dilation (3,220), clock synchronization (186,000),

Wikipedia - Gravitational time dilation

Relativity in the Global Positioning System

( ) ( ) 2 2V- 2 2V- 2 2 2 2 2 2 − ds = − 1 + c2 (cdt) + 1 − c2 (dr + r dθ + r sin θdφ ). (12 )

[ ( )2 ] V = − GME--- 1 − J a1- P (cos θ) . (13 ) r 2 r 2

′ ′ ′ ′ ′ t = t, r = r, θ = θ, φ = φ + ωEt . (14 )

⌊ ⌋ 2V ( ω r′sin θ′)2 − ds2 = − ⌈1 + --- − -E-------- ⌉ (cdt′)2 + 2ωEr ′2sin2θ ′dφ ′dt′ c2 c ( 2V ) + 1 − -2- (dr ′2 + r′2dθ′2 + r ′2sin2θ′dφ′2). (15 ) c

⌊ ⌋ 2V ( ω r′sinθ′)2 ( 2Φ ) g′00 = − ⌈1 + -2- − -E-------- ⌉ ≡ − 1 + -2- , (16 ) c c c

Φ0- GME--- GMEJ2--- ω2Ea21 c2 = − a c2 − 2a c2 − 2c2 1 1− 10 −13 −12 = − 6.95348 × 10 − 3.764 × 10 − 1.203 × 10 = − 6.96927 × 10− 10. (18 )

( ) − ds2 = − 1 + 2(Φ-−-Φ0-) (cdt′′)2 + 2ω r′2 sin2 θ′dφ ′dt′′ c2 E ( 2V ) + 1 − -2- (dr′2 + r′2d θ′2 + r′2 sin2 θ′dφ ′2), (22 ) c

( 2 (V − Φ0 )) ( 2V ) − ds2 = − 1 + -----2---- (cdt′′)2 + 1 − --2 (dr2 + r2dθ2 + r2sin2θd φ2). (23 ) c c

[ ( ) 2 2 2 2 2 2] 2 2(V--−-Φ0-) 2V- dr--+-r-dθ--+-r-sin-θd-φ- 2 − ds = − 1 + c2 − 1 − c2 (cdt)2 (cdt) . (25 )

2 2 2 2 2 2 v2 = dr--+-r-dθ-+--r-sin-θdφ--. (26 ) dt2

[ (V − Φ ) v2 ] d τ = ds∕c = 1 + -----2-0- − --2 dt. (27 ) c 2c

Terms: gravitational frequency shift (146), gravitational frequency (334),

Terms: gravitational redshift (26,400), gravitational time (8,730), redshift (1,180,000), red-shift (539,000),

Terms: time dilation (237,000), proper time (1,080,000),

Terms: general relativistic effects (1,630), accelerated reference frame (626), relativistic effects (104,000), accelerated reference frames (815), accelerated reference (1,670),

Boolean: "general relativistic" AND redshift (2,960), "general relativistic" AND "red-shift" (504), "time-dilation" AND metric (25,400), "gravitational potential" AND time (126,000),

Terms: Pound-Rebka (765),

Terms: self-referenced mode-locked lasers (13), frequency comb (14,700), femtosecond frequency combs (397), optical clock systems (277), molecular iodine clock (47), frequency combs (4,920), optical frequency combs (1,070), optical interconnects (149,000),

Terms: atomic fountain (1,770), atomic fountains (442),

Boolean: GPS AND "gravitational potential" (3,160),

The Effect of Solar Gravitational Potential. On GPS Clocks. Tom Van Flandern & Thomas B 

\sqrt{1 - v^2/c^2}

T_d = \sqrt{1 - r^2 \omega^2/c^2}

Td = 1 - gh / c2