Undergrad Gravitational Redshift: Derivation from Static Metric

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SUMMARY

The discussion focuses on deriving gravitational redshift from a static metric without relying on the equivalence principle or heuristic Newtonian methods. The key approach involves solving for a null geodesic, specifically comparing proper time along worldlines at constant radii, \(R\) (reception) and \(r\) (emission). The relationship between energy at infinity and energy measured by static observers is also explored, emphasizing the connection between proper time changes and frequency shifts, crucial for understanding redshift phenomena.

PREREQUISITES
  • Understanding of general relativity concepts, particularly null geodesics
  • Familiarity with proper time calculations in curved spacetime
  • Knowledge of gravitational redshift and its implications
  • Basic principles of Fourier analysis related to time and frequency
NEXT STEPS
  • Study the derivation of gravitational redshift using static metrics in general relativity
  • Research the mathematical formulation of null geodesics in curved spacetime
  • Explore the relationship between energy measurements at different radii in gravitational fields
  • Investigate the implications of Fourier analysis in the context of general relativity
USEFUL FOR

Physicists, astrophysicists, and students of general relativity seeking to deepen their understanding of gravitational redshift and its derivation from static metrics.

redtree
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I am trying to find a derivation of gravitational redshift from a static metric that does not depend on the equivalence principle and is not a heuristic Newtonian derivation. Any suggestions?
 
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Solve for a null geodesic. Start one at ##(r,t_1)## and the other at ##(r,t_1+\Delta t)##. Determine the time between the events where the two geodesics cross some radius ##R##?
 
Ibix said:
Determine the time

More precisely, the proper time along a worldline of constant radius ##R## (the radius of reception), as compared with the proper time along a worldline of constant radius ##r## (the radius of emission).

Another method would be to compute the energy at infinity of a null geodesic, and then show how that relates to its energy as measured by static observers at ##r## and ##R##.
 
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Given that time and frequency are Fourier conjugates, how can changes in proper time relate directly to changes in frequency (as measured by redshift)? Or am I imposing a quantum perspective on a non-quantum theory?
 
redtree said:
Given that time and frequency are Fourier conjugates, how can changes in proper time relate directly to changes in frequency (as measured by redshift)? Or am I imposing a quantum perspective on a non-quantum theory?
You are.

To get the time dilation, all you need is the proper time between emission of successive peaks at the source and the proper time between the arrival of these peaks at the destination. The ratio between the two is the time dilation factor. You could do this calculation with two flashes of light emitted an hour apart at the source if you wanted.
 

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In an inertial frame of reference (IFR), there are two fixed points, A and B, which share an entangled state $$ \frac{1}{\sqrt{2}}(|0>_A|1>_B+|1>_A|0>_B) $$ At point A, a measurement is made. The state then collapses to $$ |a>_A|b>_B, \{a,b\}=\{0,1\} $$ We assume that A has the state ##|a>_A## and B has ##|b>_B## simultaneously, i.e., when their synchronized clocks both read time T However, in other inertial frames, due to the relativity of simultaneity, the moment when B has ##|b>_B##...
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