Cosmic redshift as a purely kinematic effect

In cosmic redshift the frequency of the photon or graviton decreases and wave length increases with distance from the source. E= h×f implies that also the energy decreases. This is what Hubble's law says.

The redshift looks somewhat problematic prediction since it seems to be in conflict with the conservation of energy and momentum unless there exists a mechanism which explains how massless particle loses its four momentum gradually. In general relativity one might accept this since the very notions of energy and momentum become ill-defined. In TGD the situation is different. Four-momentum is conserved and one must understand how cosmic redshift manages to be consistent with this.

One must be very careful here since the values of wavelength and frequency, and four-momentum depend on the state of motion for the measuring system relative to the emitting system.

1. Consider first empty Minkowski space. If the measuring system is in motion with respect to source with constant velocity, one obtains a redshift or blueshift depending on the velocity and direction of the motion. The Lorentz boost inducing the shift relates different orientations of time axis for the source and receiver. The 3-D tangent space of time= constant section of receiver is obtained by Lorentz boost from that for source and the velocity associated with this boost determines the redshift.
   
   
2. Consider now what happens if time=constant slice is replaced with the cosmic time=constant slice of future directed light-cone. This surface is hyperboloid and mathematically equivalent with cosmic time=constant section of empty Robertson-Walker cosmology predicting cosmic redshift. The cosmic redshift can be understood using very simple argument. The redshift is determined by the relative orientations of the 3-D tangent spaces of hyperboloid at the positions of the source and receiver at the hyperboloid. The Lorentz boost relating the orientations of 3-D tangent spaces depends on the mutual distance and gives the cosmic redshift in empty Robertson-Walker cosmology defined by future light-cone. Cosmic redshift is mostly purely kinematical effect present already in empty space.
   
   
3. Consider next 4-D space-time surface so that also the small effects of matter are included. The time coordinate of the space-time surface could be anything but in zero energy ontology, where space-time surfaces are inside causal diamonds (intersections of future and past directed lightcones) the light-cone proper time associated with either light-like boundary of CD is the natural time coordinate defining cosmic time. Now one considers the M⁴ projections of the 3-D tangent space of the 3-D time= constant section of the space-time surface and the Lorentz boost relating the 3-D projections for source and receiver determines the redshift. The non-flatness of space-time surface gives some small corrections to the formula obtained for the case of future light-cone.
   
   
4. What about the energy of photon or graviton? Does it decrease with the distance from the sender as E=h× f requires? And is this consistent with the conservation of energy and momentum? In General Relativity one would of course have a serious problem and GRT indeed forces to give up conservation laws of four-momentum and angular momentum in long scales since these notions are not even well-defined. This was the basic motivation for TGD.
   

   In TGD there is no need to give up the conservation laws since the reduction of four-momentum predicted by redshift formula is a purely kinematic effect since the tangent spaces of sender and receiver corresponds to different states of motion.
   

The motivation for writing this comment came from proposals that cosmic redshift could be understood in terms of Planck constant, which would decrease with time in such a manner that that E=h×f would remain constant and energy would therefore be conserved despite the fact that frequency decreases (see this). At first the idea looks attractive in TGD framework, where energy is conserved. The purely kinematical explanation is however exponentially more feasible.

I have however proposed the possibility that at least part of gravitons are dark as one implication of h_gr = GMm/v₀ hypothesis. The gravitons with given frequency would have much larger energies than expected and this might dramatically affect the detection of gravitational radiation since the dark gravitons could be sees as bunches of ordinary gravitons carrying much larger energy than ordinary gravitons and could be discarded as noise. If most of gravitons are dark, the experiments trying to detect gravitons might give null results.