________________________________ From: Wouter Meeussen <wouter.meeussen@telenet.be> To: math-fun <math-fun@mailman.xmission.com> Sent: Sunday, February 9, 2014 12:40 PM Subject: [math-fun] time dilation
in http://www.nist.gov/pml/div688/2013_1_17_newera_atomicclocks_3.cfm a atomic clock, using Y-Sr ‘lattice’, is said to achieve 3E-18 accuracy & stability. This would allow to detect time dilation caused by a mere 4 cm altitude difference.
If guestimating correctly, this would be equivalent to the gravitational effect of a 19 kg mass at a horizontal distance of 10 cm.
Question: would a second 19 kg mass on the opposite side anihilate the effect (null net acceleration) or double it ? Is gravity a normal ‘garden&kitchen’ vectorial field ?
Wouter. _______________________________________________
While the electromagnetic field couples to a 4-vector, the charge density and current density, the gravitational field couples to a 16-component symmetric tensor. The tensor components are energy density, energy flux (power flow), momentum density, and momentum flux (stress). For slowly moving bodies (v << c), and in the case of gravity, not too deep a potential (GM/r << c^2), we have the force laws of Coulomb and Newton.
The gravitational redshift depends on the potential -GM/r, not the acceleration GM/r^2. The presence of the second mass doubles the potential, and thus doubles the time dilation, assuming that the test clock, the masses, and the reference clock far from the masses are all located at the same height on Earth's geoid (equipotential surface). The amount of time dilation, relative to a clock at infinity, is sqrt(1 - 2 G M / r c^2), the lower clock appearing to run slower than the upper clock.
-- Gene