Relativistic energy-loss

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Relativistic energy-loss


The choice of relativistic equations (x=0.5 or x=1) has consequences for energy-conservation.

The "bead in a box" experiment

If we equip a box with a laser source at one side and a detector at the other, and place a glass bead directly in the beam, then if the bead has velocity [math]v[/math] meters per second along the line of the beam, the final frequency [math]/nu'[/math]registering on the detector will be the original frequency, multiplied by both Doppler shifts that the signal undergoes during its journey.

Since the beam is receding from one side of the box and approaching the other, both these Doppler shifts are calculated using the same nominal velocity value but with different polarities (+/-) - one is a redshift, the other a blueshift.

... under special relativity (x=0.5)

With x=0.5 as our Lorentzlike exponent, the Doppler-shift prediction for a receding or approaching object, with [math]v[/math] quoted as recession velocity, is:

[math] \frac{\nu'}{\nu} = \sqrt{\frac{c-v}{c+v}}[/math]

This equation has the special property that when we reverse the "sign" of the velocity sign (approach<->recession), the relationship inverts exactly. The total shift after the two transitions is therefore simply:

[math] \frac{\nu}{\nu'} = \sqrt{\frac{c-v}{c+v}} × \sqrt{\frac{c+v}{c-v}} =1[/math]

The light reaches the far side of the box with no energy-change at all. This is the correct result for special relativity, which requires the presence or motion of observers or observed objects in a signal path to have zero effect on the propagation of light, or on the underlying geometry of spacetime. The light reaches the far side of the box exactly as it would have done if the bead was not there.

Under advanced GR (x=1)

In the extremal gravitoelectromagnetic solution, x=1, the result becomes:

[math] \frac{\nu}{\nu} = \frac{c-v}{c} × \frac{c-(-v)}{c} = \frac{c^2-v^2}{c^2} = 1 - \frac{v^2}{c^2}[/math]

In this case, the relative motion between bad and box changes the light-beam geometry of the box interior, so that the light arrives at its destination with less energy than it started with. The amount of energy-loss corresponds to what we would predict if each individual transition had redshifted the signal by the Lorentz factor.

Experimental consequences

(1): thermal redshifts

For x=1, if we replace the glass bead with a large number of beads bouncing about inside the box, or with the atoms of a hot gas, we will expect a light-signal passed through the region to emerge at the far side with an overall redshift.

Although this seems at first sight to be a "wrong" result, real thermal redshifts were measured for the first time in 1959, in the Pound-Rebka-Snider gravity-shift test, where it was found that the atoms in the Fe^57 crystalline lattice used as a reference had a redshift that was a function of temperature.

This thermal redshift effect had apparently not been previously predicted, and its discovery retrospectively invalidated all earlier attempts to measure the gravitational redshift effect in starlight, as none of these experiments had calculated and taken into account the thermal redshift effect in the hot stellar atmospheres. Although the experimenters are supposed to have produced a belated retrospective argument for how the shift might be arrived at using special relativity and some complex statistics, its working is not universally agreed to be valid (however, lack of unanimity over the SOD calculations didn't affect the validity of the experiment's central gravity-shift calculations).

(2): cosmological redshifts

A larger-scale consequence of relativistic energy-loss arises in the case of gravitational shifts. If the gravitational shift on light crossing a differential with terminal velocity [math]v[/math] changes energy according to the SR Doppler relationship, then we might expect the light to regain all its original energy when it leaves the region and returns to its initial location. we would then expect light skimming a star, or reflecting off a mirror placed in a gravity-well, to not undergo any overall redshift over the round trip.

With the redder equations of x=1 (or any other intermediate relationships in the range 0.5<x<=1), we expect a net energy-loss. if we now consider the large-scale structure of the universe, where a signal may have to ride a switchback" sequence of gravitational redshifts and blueshifts before it reaches us, , we should, if the distribution of stars in the universe was totally uniform, expect light from distant sources to show an energy-loss (on average) proportional to the distance travelled.

It is important to note that in the advanced general theory, this effect is not a competing explanation of Hubble shift, but a dual description, with the universal shift equivalence principle that appears in AGR, we can standardise on x=1, predict a cosmological redshift, invoke recession-equivalence, and then use this to argue for the necessity of an expanding universe.


Although at first sight it might seem that violating energy-conservation automatically rules out the x=1 solution, the two main classes of "violating" behaviour that it predicts have been confirmed by experiment.

See also

  • cosmological shift equivalence
  • Advanced General Relativity.

Further reading

  • pound-snider
  • Pound-snider2

Eric Baird , , , ' , ' , Relativity in Curved Spacetime , ' ' pages ' (2007)

section xxx


  • The unanticipated thermal redshift effect that confounded the experimenters' initial attempts to measure gravitational shifts was referred to rather colourfully in the 1959 paper as the "SOD" effect (for "Second Order Doppler").
  • If we had been using an x&equals;1 theory rather than special relativity, we should have been able to predict the existence of thermal redshifts in advance, rather than stumbling across the effect by accident.
  • If this form of energy-loss invalidates a theory, then (thanks to Hubble redshift) pretty much every current mainstream cosmological theory and model, including ordinary GR, would have to be considered invalidated, too.