Cosmological Redshift in FRW Coordinates – Energy Conservation & Time Dilation

[nutgeb]

I want to discuss energy conservation and time dilation relative to the cosmological redshift. Because the elements of the redshift are coordinate specific, I’ll focus just on the FRW metric using proper distance coordinates (not comoving coordinates).

As explained in Tamara Davis’s cover article in the July issue of Scientific American, and by Bunn & Hogg in a paper she references, in FRW coordinates the cosmological redshift can be thought of as photons experiencing Doppler shifts along their trajectory. This interpretation assumes that comoving galaxies move through space, and are not dragged apart by the expansion of space itself.

In Minkowski coordinates, Doppler shift arises from a single end-to-end velocity differential between the emitter and observer. But in FRW coordinates, Doppler shift occurs in many tiny increments all along the trajectory. Relative to the emitter’s reference frame, each tiny segment of a photon’s worldline is characterized by a higher proper recession velocity than the previous segment the photon passed through. This recession velocity differential is a feature of Hubble’s Law: proper recession velocity increases in direct proportion to the proper distance from the emitter. The Hubble velocity in each segment is HD, where H is the current Hubble rate and D is the proper distance.

The photon is constrained to always maintain a local peculiar velocity of exactly c. So its proper velocity relative to the emitter starts simply as V = c, but thereafter is progressively boosted by the increasing Hubble velocities it passes through: V = c + HD (where V is proper velocity relative to the emitter). Because we are using FRW coordinates, there is no problem with the proper velocity exceeding c, just as the Hubble velocity itself can exceed c.

The Doppler shift occurs because, when light crosses into a segment of the worldline which has a faster Hubble recession velocity than the previous segment, the arrival time between each successive wave crest is slightly delayed. Relative to any given reference frame, the light waves progressively stretch out as they travel, because an earlier wave crest is always slightly farther away from the emitter than a subsequent wave crest, and therefore is constrained to have a higher proper velocity away from the emitter than the subsequent wave crest does at each instant in time. So the proper distance between any two wave crests progressively increases as a function of proper distance from the emitter (and to a lesser extent also as a function of the ever-changing Hubble rate). This is why each increment of redshift is fully realized immediately en route, rather than being realized entirely only upon arrival at the observer, even though the photon is not observed in the intermediate segments it passes through. The amount of redshift would be very different if calculated as a single end-to-end event instead of as an accumulation of progressive increments.

Energy conservation

I would like to discuss how the redshift conserves energy. The Davis article points out that the locally measured energy of a photon progressively decays as it redshifts incrementally while moving from frame to frame along its trajectory. But she concludes that, like all Doppler shifts, the apparent loss of energy merely reflects a change in perspective, that is, a change in reference frame, rather than a violation of energy conservation. The universe therefore is not 'leaking energy' as a consequence of redshift.

It is misleading to focus only on standalone local energy measurements. When the emitter and observer are in motion relative to each other, energy conservation is meaningful only if each emitter-observer pair is considered in isolation, and then the results are aggregated for the universe as a whole. Energy is strictly conserved if one takes the Doppler velocity differentials along the photon’s worldline fully into account.

Cosmological Doppler shifts can be thought of as momentum decay that is required by energy conservation. A photon’s momentum decays as it redshifts, in proportion to the decay in its frequency.

As I described, the photon gains proper velocity relative to the emitter along its trajectory. An increase in proper velocity does not come as a free lunch; for the photon to avoid gaining energy relative to the emitter's reference frame, its momentum must decay by the same ratio as its proper velocity increases. That is exactly what occurs. Proper velocity increases in proportion to the expansion of the cosmic scale factor a, while momentum decays by the inverse proportion, 1/a.

At each location along the photon's trajectory, multiplying the magnitude of the momentum p and proper velocity V together always yields a constant. I think that constant relates the photon's energy conservation back to the emitter's reference frame. Photon energy normally is calculated as pc. The photon’s proper velocity is c at the instant of emission, but thereafter as it gains proper velocity, the equation pc calculates only the locally measured energy. In order to relate the energy back to the emitter’s frame, momentum must be multiplied by proper velocity V rather than local velocity c: E = pV.

Doppler shift equation and time dilation

Mathematically, the incremental Doppler shifts are multiplied together to calculate the total accumulated redshift. The equation for this is the multiplicative series (1+H_t1dD_t1)(1+H_t2dD_t2)...(1+H_t0dD_t0). Here dD is the change in proper distance from the emitter that occurs within each tiny segment of the worldline, and t0 is the time of observation. So H_tdD simply calculates the Hubble recession velocity differential, and therefore the amount of Doppler shift, that the photon experiences when it crosses one segment.

Note that for each segment, dD = dt if units of c = 1 are used for time and distance. H_tdt calculates how much the scale factor of the universe has expanded during the time duration of one segment of the photon’s travel. These also are aggregated together multiplicatively to calculate the total expansion. The fact that H_tdt = H_tdD demonstrates unequivocally that the kinematic Doppler solution is exactly equal to the traditional cosmological redshift formula whereby the redshift is equal to the proportional amount by which the cosmic scale factor has expanded during the photon’s travel. Thus, neither observations nor math provide a basis for preferring either the kinematic paradigm or the expanding space paradigm for the expansion of the universe.

It is very interesting that the cosmological Doppler shift formula in FRW coordinates is in the same form as the classical (non-relativistic) Doppler shift formula for a moving emitter and stationary observer: 1 + V/c. This shows that the cosmological Doppler shift, when calculated in FRW coordinates, does not include any element of SR time dilation. This is because of a characteristic unique to the FRW metric -- no time dilation occurs between comovers despite their unbounded recession velocities relative to each other. The clocks of all comovers run at the same rate and record the same elapsed time since the Big Bang.

To illustrate this point, consider the empty Milne model. In that case, SR Minkowski coordinates are converted to FRW coordinates by applying a Lorentz length expansion gamma factor 1/$$\gamma$$ (see Peacock p. 88), which offsets the SR time dilation gamma factor $$\gamma$$ for comovers (but not for peculiar velocities). Compared to the SR velocity in Minkowski coordinates, the GR velocity in FRW coordinates is increased by the factor arctanh:

$$V_{FRW} = \frac{1}{2} ln\left( \frac{1 + v_{sr} }{ 1 - v_{sr} } \right)$$

Also consider that in FRW coordinates Hubble comoving recession velocities are always added together directly without using the SR formula for addition of velocities. The SR formula obviously wouldn’t work because Hubble velocities can exceed c.

I would like to discuss this point. You may think that a Doppler equation that does not include the effects of SR is flawed, but consider whether that logic applies specifically to comovers in FRW coordinates. You may prefer to think of this equation as being equivalent to the SR Doppler shift, but with the time dilation element eliminated due to the use of FRW coordinates. The element of SR time dilation is the only difference between the classical and SR Doppler shifts.

I should emphasize that as the segment lengths are reduced toward zero, the SR time dilation element approaches zero anyway. So as a practical matter the difference between including and omitting SR time dilation from the equation can be made vanishingly small. However it is an interesting theoretical question.

And by way of comparison, if Schwarzschild coordinates are used instead of FRW, then the redshift equation includes elements of both SR and gravitational time dilation, neither of which appears directly in the FRW version of the redshift equation.

 

[nutgeb]

Regarding time dilation, I'll also point out that the two axes used in FRW proper distance coordinates are cosmic proper time and proper distance -- unlike Minkowski coordinates where the axes are set to local coordinate time and local coordinate distance based on a single inertial reference frame. Therefore, FRW charts reflect the relative velocities of fundamental comoving objects in a more homogeneous way than Minkowski coordinates do. The proper time and distance axes capture the local velocities of all comovers, without the kind of Lorentz Boost vector "rotation" that is portrayed on a Minkowski spacetime chart.

The linearity of Hubble's Law is effectively "built in" to FRW coordinates. It is not possible to depict coordinate time dilation as between any two comovers on a FRW chart without changing the distribution of comovers to diverge from Hubble's Law.

As I said in my post, an inverse Lorentz transformation is "built in" to FRW coordinates. In theory, for example, FRW coordinates could be used to portray a scenario without gravity, comprised of only 2 test objects which both started at the origin (let's call it "Earth"), with one of them immediately moving away the other at a constant proper velocity. If both test objects are treated as FRW fundamental comovers (i.e., we define the time-dependent Hubble rate built into the FRW chart to align with the moving object's contemporaneous velocity) it would be perfectly permissible for that proper velocity to exceed c. In this coordinate system no time dilation or Lorentz contraction would occur as between the inertial reference frames of the two test objects. From this we learn that c is the speed limit only locally within a comoving reference frame, but not as between different comoving objects in different reference frames.

This is simply how FRW coordinates work, and it complies with GR while not violating SR. SR does not apply as a discrete element to comovers in FRW coordinates, only to peculiar velocities.

(Of course in our current universe, the Hubble rate is so small that relativistic recession velocities exist only at cosmic distances, e.g. z > .5. But in the very early universe, the Hubble rate was enormously larger.)

One might think that FRW coordinates are somehow stilted or unnatural compared to Minkowski coordinates, but I don't think so. FRW coordinates work perfectly well at distances of 1 meter or 10 Gly (there is no requirement to approximate to Minkowski coordinates locally, as is often done). FRW coordinates are the clearest coordinate system for portraying simultaneously what all comovers at different radial distances observe about each other. FRW coordinates naturally portray what a series of many such comovers would measure if they laid rulers end to end between themselves or compared clocks with their neighbors, despite the fact that their neighbors are in motion relative to them. Minkowski coordinates must go through gyrations to portray those aspects clearly. For example, a curved light cone in FRW coordinates accurately portrays the proper length of the photon's locally-measured worldline, while the straight light cone in Minkowski coordinates portrays a Lorentz contracted length that does not comply with aggregated local measurements. But on the other hand, Minkowski coordinates clearly are more easily adaptable to a broader range of scenarios.

For some reason the unusual aspects of FRW coordinates don't get much discussion (other than that they permit superluminal recession velocities). Of course since velocities are coordinate-specific there's no reason to assume that any two quite different coordinate systems would portray them the same way.

Reasoned comments and questions are encouraged.

 

[Austin0]

I want to discuss energy conservation and time dilation relative to the cosmological redshift. Because the elements of the redshift are coordinate specific, I’ll focus just on the FRW metric using proper distance coordinates (not comoving coordinates).

As explained in Tamara Davis’s cover article in the July issue of Scientific American, and by Bunn & Hogg in a paper she references, in FRW coordinates the cosmological redshift can be thought of as photons experiencing Doppler shifts along their trajectory. This interpretation assumes that comoving galaxies move through space, and are not dragged apart by the expansion of space itself.

In Minkowski coordinates, Doppler shift arises from a single end-to-end velocity differential between the emitter and observer. But in FRW coordinates, Doppler shift occurs in many tiny increments all along the trajectory. Relative to the emitter’s reference frame, each tiny segment of a photon’s worldline is characterized by a higher proper recession velocity than the previous segment the photon passed through. This recession velocity differential is a feature of Hubble’s Law: proper recession velocity increases in direct proportion to the proper distance from the emitter. The Hubble velocity in each segment is HD, where H is the current Hubble rate and D is the proper distance.

The photon is constrained to always maintain a local peculiar velocity of exactly c. So its proper velocity relative to the emitter starts simply as V = c, but thereafter is progressively boosted by the increasing Hubble velocities it passes through: V = c + HD (where V is proper velocity relative to the emitter). Because we are using FRW coordinates, there is no problem with the proper velocity exceeding c, just as the Hubble velocity itself can exceed c.

The Doppler shift occurs because, when light crosses into a segment of the worldline which has a faster Hubble recession velocity than the previous segment, the arrival time between each successive wave crest is slightly delayed. Relative to any given reference frame, the light waves progressively stretch out as they travel, because an earlier wave crest is always slightly farther away from the emitter than a subsequent wave crest, and therefore is constrained to have a higher proper velocity away from the emitter than the subsequent wave crest does at each instant in time. So the proper distance between any two wave crests progressively increases as a function of proper distance from the emitter (and to a lesser extent also as a function of the ever-changing Hubble rate). This is why each increment of redshift is fully realized immediately en route, rather than being realized entirely only upon arrival at the observer, even though the photon is not observed in the intermediate segments it passes through. The amount of redshift would be very different if calculated as a single end-to-end event instead of as an accumulation of progressive increments.

Energy conservation

I would like to discuss how the redshift conserves energy. The Davis article points out that the locally measured energy of a photon progressively decays as it redshifts incrementally while moving from frame to frame along its trajectory. But she concludes that, like all Doppler shifts, the apparent loss of energy merely reflects a change in perspective, that is, a change in reference frame, rather than a violation of energy conservation. The universe therefore is not 'leaking energy' as a consequence of redshift.

It is misleading to focus only on standalone local energy measurements. When the emitter and observer are in motion relative to each other, energy conservation is meaningful only if each emitter-observer pair is considered in isolation, and then the results are aggregated for the universe as a whole. Energy is strictly conserved if one takes the Doppler velocity differentials along the photon’s worldline fully into account.

Cosmological Doppler shifts can be thought of as momentum decay that is required by energy conservation. A photon’s momentum decays as it redshifts, in proportion to the decay in its frequency.

As I described, the photon gains proper velocity relative to the emitter along its trajectory. An increase in proper velocity does not come as a free lunch; for the photon to avoid gaining energy relative to the emitter's reference frame, its momentum must decay by the same ratio as its proper velocity increases. That is exactly what occurs. Proper velocity increases in proportion to the expansion of the cosmic scale factor a, while momentum decays by the inverse proportion, 1/a.

At each location along the photon's trajectory, multiplying the magnitude of the momentum p and proper velocity V together always yields a constant. I think that constant relates the photon's energy conservation back to the emitter's reference frame. Photon energy normally is calculated as pc. The photon’s proper velocity is c at the instant of emission, but thereafter as it gains proper velocity, the equation pc calculates only the locally measured energy. In order to relate the energy back to the emitter’s frame, momentum must be multiplied by proper velocity V rather than local velocity c: E = pV.

Doppler shift equation and time dilation

Mathematically, the incremental Doppler shifts are multiplied together to calculate the total accumulated redshift. The equation for this is the multiplicative series (1+H_t1dD_t1)(1+H_t2dD_t2)...(1+H_t0dD_t0). Here dD is the change in proper distance from the emitter that occurs within each tiny segment of the worldline, and t0 is the time of observation. So H_tdD simply calculates the Hubble recession velocity differential, and therefore the amount of Doppler shift, that the photon experiences when it crosses one segment.

Note that for each segment, dD = dt if units of c = 1 are used for time and distance. H_tdt calculates how much the scale factor of the universe has expanded during the time duration of one segment of the photon’s travel. These also are aggregated together multiplicatively to calculate the total expansion. The fact that H_tdt = H_tdD demonstrates unequivocally that the kinematic Doppler solution is exactly equal to the traditional cosmological redshift formula whereby the redshift is equal to the proportional amount by which the cosmic scale factor has expanded during the photon’s travel. Thus, neither observations nor math provide a basis for preferring either the kinematic paradigm or the expanding space paradigm for the expansion of the universe.

It is very interesting that the cosmological Doppler shift formula in FRW coordinates is in the same form as the classical (non-relativistic) Doppler shift formula for a moving emitter and stationary observer: 1 + V/c. This shows that the cosmological Doppler shift, when calculated in FRW coordinates, does not include any element of SR time dilation. This is because of a characteristic unique to the FRW metric -- no time dilation occurs between comovers despite their unbounded recession velocities relative to each other. The clocks of all comovers run at the same rate and record the same elapsed time since the Big Bang.

To illustrate this point, consider the empty Milne model. In that case, SR Minkowski coordinates are converted to FRW coordinates by applying a Lorentz length expansion gamma factor 1/$$\gamma$$ (see Peacock p. 88), which offsets the SR time dilation gamma factor $$\gamma$$ for comovers (but not for peculiar velocities). Compared to the SR velocity in Minkowski coordinates, the GR velocity in FRW coordinates is increased by the factor arctanh:

$$V_{FRW} = \frac{1}{2} ln\left( \frac{1 + v_{sr} }{ 1 - v_{sr} } \right)$$

Also consider that in FRW coordinates Hubble comoving recession velocities are always added together directly without using the SR formula for addition of velocities. The SR formula obviously wouldn’t work because Hubble velocities can exceed c.

I would like to discuss this point. You may think that a Doppler equation that does not include the effects of SR is flawed, but consider whether that logic applies specifically to comovers in FRW coordinates. You may prefer to think of this equation as being equivalent to the SR Doppler shift, but with the time dilation element eliminated due to the use of FRW coordinates. The element of SR time dilation is the only difference between the classical and SR Doppler shifts.

I should emphasize that as the segment lengths are reduced toward zero, the SR time dilation element approaches zero anyway. So as a practical matter the difference between including and omitting SR time dilation from the equation can be made vanishingly small. However it is an interesting theoretical question.

.

OK the reciprocally variable momentum and velocity neatly does away with the conservation problem in flight but...

If you assume that the velocity increases with distance from the source this would apply locally for any small interval along the path but what is the justification for any assumption of accumulation?

It seems that it would only be a transitory relationship.

If you assume that the velocity locally at emission and at reception is equivalent it would seem to follow that the increased velocity from origen must undergo an equivalent reduction entering the gravitational approach to the destination, with an equivalent contraction of the wavelength. Unless it somehow instantly reduces speed at reception without affecting wavelength.

You mentioned dilation pertaining to Doppler in SR.

My understanding was that Doppler in SR is based on relative motion of the emitter and receptor. Basically transforming the classical wave equations into Minkowski. With no implication of dilation of the periodicity of the emitter or reception electrons involved.
AM I mistaken?

 

[nutgeb]

If you assume that the velocity increases with distance from the source this would apply locally for any small interval along the path but what is the justification for any assumption of accumulation?

It seems that it would only be a transitory relationship.

Thanks for the question.

The photon's proper velocity continues to increase in proportion to distance from the emitter, because it is dictated by Hubble's Law, which continues to increase in proportion to distance. The photon's proper velocity is V = c + HD. Obviously the increases in proper recession velocity in each successive segment of the photon's worldline away from the emitter will accumulate and they are not transitory.

If you assume that the velocity locally at emission and at reception is equivalent it would seem to follow that the increased velocity from origen must undergo an equivalent reduction entering the gravitational approach to the destination, with an equivalent contraction of the wavelength. Unless it somehow instantly reduces speed at reception without affecting wavelength.

No, historically a photon's proper velocity relative to the emitter has not undergone any reduction at any time up to the instant of impact at the observer. You must remember that the cosmological redshift is determined from the Hubble velocity HD, not from the Hubble rate H alone. Cosmological redshift historically has continued to accumulate en route even though the end-to-end proper recession velocity as between the emitter and observer has decreased for most of the history of the universe (until dark energy began to dominate over gravity, reaccelerating the expansion).

You mentioned the effect of gravity on the photon. Of course gravitational acceleration is coordinate-dependent. In FRW coordinates, gravity is not a discrete element of the Doppler shift equation which calculates the redshift, because the emitter and observer are in freefall relative to each other. Instead, gravity's role is indirect: it progressively slows the cosmic Hubble rate over time. In this way, gravity does affect the photon's increase in proper velocity through the equation V = c + HD, by making the proper velocity increase during each segment of the worldline less than it would be if there were no gravity. And therefore there is less total redshift than there would have been without gravity. But historically the photon's total accumulated proper velocity has always increased as a function of elapsed time en route, because the incremental proportional increase in proper distance D from the emitter has always exceeded the contemporaneous incremental proportional decay in the Hubble rate H.

(By the way, the Hubble rate would decrease as a function of time even if there were no gravity (and no dark energy). The Hubble rate is stated in units of absolute distance (e.g., km/sec/Mpc), so if the proper recession velocity between two comoving galaxies remains constant, mathematically the Hubble rate must decrease as the proper distance between them increases. Gravity just adds to the rate of decay.)

My understanding was that Doppler in SR is based on relative motion of the emitter and receptor. Basically transforming the classical wave equations into Minkowski. With no implication of dilation of the periodicity of the emitter or reception electrons involved.
AM I mistaken?

SR Doppler shift can be decomposed into two elements:

First, the classical Doppler shift element, in which the photon's wavelength is stretched in the observer's frame because the emitter is in motion relative to the observer. This element is z + 1 = 1+v/c for a stationary observer and moving emitter.

Second, the SR time dilation factor, in which the photon's wavelength is observed to be longer in the observer's frame (than if observed in the emitter's frame) to an additional degree, because of the SR time dilation resulting from the emitter and observer being in motion relative to each other at a relativistic velocity. In Minkowski coordinates this element is just the gamma factor $$\gamma$$ :

$$\frac{1}{\sqrt{ (1+v/c) (1-v/c) } }$$

In Minkowski coordinates, both elements, multiplied together, apply to yield the total SR Doppler shift:

$$z + 1 = \sqrt{ \frac{1+v/c} {1-v/c} }$$

But in FRW coordinates, the relative recession motion between a comoving emitter and observer does not result in time dilation, and their clocks run at the same rate. Therefore only the "classical" element applies, as I explained. Again this is just a coordinate-specific outcome.

 

[Austin0]

Thanks for the question.

The photon's proper velocity continues to increase in proportion to distance from the emitter, because it is dictated by Hubble's Law, which continues to increase in proportion to distance. The photon's proper velocity is V = c + HD. Obviously the increases in proper recession velocity in each successive segment of the photon's worldline away from the emitter will accumulate and they are not transitory.

No, historically a photon's proper velocity relative to the emitter has not undergone any reduction at any time up to the instant of impact at the observer. You must remember that the cosmological redshift is determined from the Hubble velocity HD, not from the Hubble rate H alone. Cosmological redshift historically has continued to accumulate en route even though the end-to-end proper recession velocity as between the emitter and observer has decreased for most of the history of the universe (until dark energy began to dominate over gravity, reaccelerating the expansion).

You mentioned the effect of gravity on the photon. Of course gravitational acceleration is coordinate-dependent. In FRW coordinates, gravity is not a discrete element of the Doppler shift equation which calculates the redshift, because the emitter and observer are in freefall relative to each other. Instead, gravity's role is indirect: it progressively slows the cosmic Hubble rate over time. In this way, gravity does affect the photon's increase in proper velocity through the equation V = c + HD, by making the proper velocity increase during each segment of the worldline less than it would be if there were no gravity. And therefore there is less total redshift than there would have been without gravity. But historically the photon's total accumulated proper velocity has always increased as a function of elapsed time en route, because the incremental proportional increase in proper distance D from the emitter has always exceeded the contemporaneous incremental proportional decay in the Hubble rate H.

(By the way, the Hubble rate would decrease as a function of time even if there were no gravity (and no dark energy). The Hubble rate is stated in units of absolute distance (e.g., km/sec/Mpc), so if the proper recession velocity between two comoving galaxies remains constant, mathematically the Hubble rate must decrease as the proper distance between them increases. Gravity just adds to the rate of decay.)

SR Doppler shift can be decomposed into two elements:

First, the classical Doppler shift element, in which the photon's wavelength is stretched in the observer's frame because the emitter is in motion relative to the observer. This element is z + 1 = 1+v/c for a stationary observer and moving emitter.

Second, the SR time dilation factor, in which the photon's wavelength is observed to be longer in the observer's frame (than if observed in the emitter's frame) to an additional degree, because of the SR time dilation resulting from the emitter and observer being in motion relative to each other at a relativistic velocity. In Minkowski coordinates this element is just the gamma factor $$\gamma$$ :

$$\frac{1}{\sqrt{ (1+v/c) (1-v/c) } }$$

In Minkowski coordinates, both elements, multiplied together, apply to yield the total SR Doppler shift:

$$z + 1 = \sqrt{ \frac{1+v/c} {1-v/c} }$$

But in FRW coordinates, the relative recession motion between a comoving emitter and observer does not result in time dilation, and their clocks run at the same rate. Therefore only the "classical" element applies, as I explained. Again this is just a coordinate-specific outcome.

If you are supposing that the velocity increase continues cummulatively up to reception why do you think that this wouldn't raise frequency at reception. Energy, momentum etc?

I was talking about gravity slowing down the light not speeding it up.

If light speed is a function of potential radius or distance, then the intergalactic space would be the maximum speed. Coming into a lower potential approaching the receptor , galaxy, solar system and finally planet it would seem to follow that there would be some non negligable decrease

On the SR Doppler. You seemed to use the z from cosmic shift with an additional 1

I am only familiar with

[tex] E = \sqrt{ \frac{1+v/c} {1-v/c} } [/tex] or for v the same

If time dilation is assumed for the absorbing electron why is it not also for the emitting electron?

 

[nutgeb]

If you are supposing that the velocity increase continues cumulatively up to reception why do you think that this wouldn't raise frequency at reception. Energy, momentum etc?

The point I was making is that there is no free lunch when a photon's proper velocity increases relative to the emitter's frame; there is nothing contributing energy to increase the photon's frequency and therefore its total energy. The photon is "forced" to speed up to maintain a local speed of c as the Hubble velocity it passes through increases. It is not somehow "pulled", or "falling downhill". In order to conserve energy in the emitter's frame, it must lose the same proportionate amount of momentum as the proper velocity it gains. Relativistic particles like photons are different than massive particles -- a photon's local speed is fixed at c, and its locally measured momentum depends on its wavelength or frequency rather than on its local speed.

I was talking about gravity slowing down the light not speeding it up.

If light speed is a function of potential radius or distance, then the intergalactic space would be the maximum speed. Coming into a lower potential approaching the receptor , galaxy, solar system and finally planet it would seem to follow that there would be some non negligible decrease.

Something is confused here. First, the distance I was talking about is the distance from the emitter, not the observer. Second, a photon approaching a massive body such as a galaxy would be entering a higher gravitational potential, not a lower one, so from the observer's perspective it would become a bit blueshifted not redshifted. Anyway, for a photon traveling from a large cosmic distance, the already accumulated cosmological redshift would dominate over the relatively de minimus bit of blueshift caused by approaching a galaxy. My discussion of the cosmological redshift is simplified to assume a perfectly homogeneous FRW universe. The effect of a local homogeneity can be easily calculated but it's not what I'm addressing.

On the SR Doppler. You seemed to use the z from cosmic shift with an additional 1

I am only familiar with

[tex] E = \sqrt{ \frac{1+v/c} {1-v/c} } [/tex] or for v the same

The equation I gave for the SR Doppler shift is the correct one. See the Wikipedia article http://en.wikipedia.org/wiki/Relativistic_Redshift#Relativistic_effects".

If time dilation is assumed for the absorbing electron why is it not also for the emitting electron?

I think you've missed the point here. Time dilation compares the observer's clock with the emitter's clock. If the emitter's clock is running slower (as it would be if the emitter were moving away at a relativistic speed in Minkowski coordinates) then the light it emits will be observed to be more redshifted by the observer, whose clock is running relatively faster. Again, check out Wikipedia for a basic description of how it works.

 

[Austin0]

The point I was making is that there is no free lunch when a photon's proper velocity increases relative to the emitter's frame; there is nothing contributing energy to increase the photon's frequency and therefore its total energy. The photon is "forced" to speed up to maintain a local speed of c as the Hubble velocity it passes through increases. It is not somehow "pulled", or "falling downhill". In order to conserve energy in the emitter's frame, it must lose the same proportionate amount of momentum as the proper velocity it gains. Relativistic particles like photons are different than massive particles -- a photon's local speed is fixed at c, and its locally measured momentum depends on its wavelength or frequency rather than on its local speed.

Something is confused here. First, the distance I was talking about is the distance from the emitter, not the observer. Second, a photon approaching a massive body such as a galaxy would be entering a higher gravitational potential, not a lower one, so from the observer's perspective it would become a bit blueshifted not redshifted. Anyway, for a photon traveling from a large cosmic distance, the already accumulated cosmological redshift would dominate over the relatively de minimus bit of blueshift caused by approaching a galaxy. My discussion of the cosmological redshift is simplified to assume a perfectly homogeneous FRW universe. The effect of a local homogeneity can be easily calculated but it's not what I'm addressing.

The equation I gave for the SR Doppler shift is the correct one. See the Wikipedia article http://en.wikipedia.org/wiki/Relativistic_Redshift#Relativistic_effects".

I think you've missed the point here. Time dilation compares the observer's clock with the emitter's clock. If the emitter's clock is running slower (as it would be if the emitter were moving away at a relativistic speed in Minkowski coordinates) then the light it emits will be observed to be more redshifted by the observer, whose clock is running relatively faster. Again, check out Wikipedia for a basic description of how it works.

I think one of us is confused here. Gravititational potential increases with increasing distance from the mass as far as I know.

Light slows down at levels of decreased potential i.e. closer to center of mass.

This would mean that leaving a field the speed would increase and entering a field speed would decrease. Obnviously in intergalactic space the field potential would be maximal ,maximum speed.

On DOppler and dilation I think you missed my point. If you assume dilation then at the emitter the electron freqency would be lower and the photon would start out with a longer wavelength no??

You keep talking about Hubbles law and constant as a basis for assuming photon behavior in trransit but aren't both related to the relationship between galaxies with no explicit or necessary assumptions about the space in between?

 

[nutgeb]

I think one of us is confused here. Gravititational potential increases with increasing distance from the mass as far as I know.

Maybe the confusion here is that Newtonian gravitational potential is stated as a negative value. It is zero at an infinite distance and becomes increasingly negative as the central mass is approached. I was referring to the absolute value of the potential, i.e. the amount of acceleration it imparts to a freefalling particle. The absolute value of potential is highest near the mass and decreases with distance.

Light slows down at levels of decreased potential i.e. closer to center of mass.

This would mean that leaving a field the speed would increase and entering a field speed would decrease.

Obnviously in intergalactic space the field potential would be maximal ,maximum speed.

Clock rates are slower closer to a Schwarzschild gravitational mass, and relatively faster further away. Therefore a photon approaching a Schwarzschild mass is observed to be blueshifted. A photon moving away would be redshifted. But as I said, the extra bit of blueshift from approaching a local mass has a de minimus effect on the total amount of cosmological redshift from a photon traveling from a great cosmic distance.

It is way too complex to calculate whether local inhomogeneities in the matter distribution may change the FRW metric itself so as to affect the calculation of a photon's increase in proper velocity relative to the emitter's comoving frame. If that's what you are suggesting by your remarks about the speed of light changing across a gravitational potential gradient. But as I said the total (and rather de minimus) effect on the redshift can be calculated by considering the two elements separately: First the cosmological redshift due to the change in the photon's proper velocity assuming a homogeneous matter distribution. Then multiply that by the bit of blueshift that occurs because of approaching a local gravitational mass. That's about as far as I can go on this subject.

On DOppler and dilation I think you missed my point. If you assume dilation then at the emitter the electron freqency would be lower and the photon would start out with a longer wavelength no??

Depends on which reference frame you're referring to. Assuming Minkowski coordinates (which is not what this thread is about):

In the observer's rest frame, the emitter is moving way (receding), therefore per Special Relativity the emitter's clock runs slower than the observer's clock. Therefore arriving photons appear redshifted in the observer's rest frame, representing the product of classical Doppler shift and SR time dilation. The observer can extend its local inertial rest frame to encompass the distant emitter, in which case the observer can calculate that the photon had a lower frequency and greater wavelength from the instant of emission (compared to what would be calculated in the emitter's rest frame).

If the photon is instead observed in the emitter's own rest frame, then the emitter is obviously at rest relative to the observer, therefore there can be no time dilation, and no Doppler redshift at all occurs.

However, the above discussion is specific to Minkowski coordinates. In FRW coordinates, the observer cannot extend its local inertial rest frame to encompass the distant receding emitter. Instead, the FRW observer must calculate the incremental Doppler shift in each intervening receding local frame, and multiply them together to calculate the total redshift. That is a fundamental difference between Minkowski and FRW coordinates.

You keep talking about Hubbles law and constant as a basis for assuming photon behavior in trransit but aren't both related to the relationship between galaxies with no explicit or necessary assumptions about the space in between?

That would be true in Minkowski coordinates. But my OP is about FRW coordinates, so it is not true.

As I explained, in FRW coordinates Hubble's Law requires a departing photon to accumulate proper velocity increases relative to the emitter en route, regardless of whether that photon is ever observed. If the photon is eventually observed, and the observer is at rest in its local comoving FRW reference frame, then the observer's own proper recession velocity will be exactly the same as the proper velocity, minus c, that the photon had already gained (relative to the emitter) at the instant just before reception. Exactly as I described in the OP. So in the case of a purely comoving emitter and observer, since each is at rest in its own FRW comoving frame, it has no additional velocity to add to the redshift beyond the recession velocity that the comoving frame itself had, as required by Hubble's Law.

That's how FRW coordinates work -- the reference frames themselves are considered to be receding away from each other at the Hubble velocity HD, regardless of whether a particular reference frame contains an observer. Many people interpret that to mean that in FRW coordinates empty space itself is expanding and dragging the galaxies apart (like a line of dots marked on the surface of an expanding balloon), which is fine if one applies the analogy carefully, but that interpretation is not necessary to describe and calculate the observed redshift.

 

[Austin0]

I think one of us is confused here. Gravititational potential increases with increasing distance from the mass as far as I know.

Maybe the confusion here is that Newtonian gravitational potential is stated as a negative value. It is zero at an infinite distance and becomes increasingly negative as the central mass is approached. I was referring to the absolute value of the potential, i.e. the amount of acceleration it imparts to a freefalling particle. The absolute value of potential is highest near the mass and decreases with distance.

OK it seems that you are not talking about potential because as far as I know the term has not changed meaning since Newton.

Light slows down at levels of decreased potential i.e. closer to center of mass.

Clock rates are slower closer to a Schwarzschild gravitational mass, and relatively faster further away. Therefore a photon approaching a Schwarzschild mass is observed to be blueshifted. A photon moving away would be redshifted. But as I said, the extra bit of blueshift from approaching a local mass has a de minimus effect on the total amount of cosmological redshift from a photon traveling from a great cosmic distance.

This would mean that leaving a field the speed would increase and entering a field speed would decrease. Obnviously in intergalactic space the field potential would be maximal ,maximum speed.
.

It is way too complex to calculate whether local inhomogeneities in the matter distribution may change the FRW metric itself so as to affect the calculation of a photon's increase in proper velocity relative to the emitter's comoving frame. If that's what you are suggesting by your remarks about the speed of light changing across a gravitational potential gradient. But as I said the total (and rather de minimus) effect on the redshift can be calculated by considering the two elements separately: First the cosmological redshift due to the change in the photon's proper velocity assuming a homogeneous matter distribution. Then multiply that by the bit of blueshift that occurs because of approaching a local gravitational mass. That's about as far as I can go on this subject.

Well my knowledge of GR is limited to say the least but from what is being said in this forum aside from geodesic inertial effects on a photon there is a decrease in actual speed with decreasing Schwarzschild radius. This may have negligable local effect but if true would seem to mean significant effect between the intergalactic space and the densely populated galactic space and then the less dense but shorter radius of the solar system and earth.

I am not prepared to argue this point but it seems like it could be too relevant a factor to simply dismiss it. I don't mean you personally but whoever's hypotheses this is.

You keep talking about Hubbles law and constant as a basis for assuming photon behavior in trransit but aren't both related to the relationship between galaxies with no explicit or necessary assumptions about the space in between?

As I explained, in FRW coordinates Hubble's Law requires a departing photon to accumulate proper velocity increases relative to the emitter en route, regardless of whether that photon is ever observed. If the photon is eventually observed, and the observer is at rest in its local comoving FRW reference frame, then the observer's own proper recession velocity will be exactly the same as the proper velocity, minus c, that the photon had already gained (relative to the emitter) at the instant just before reception. Exactly as I described in the OP. So in the case of a purely comoving emitter and observer, since each is at rest in its own FRW comoving frame, it has no additional velocity to add to the redshift beyond the recession velocity that the comoving frame itself had, as required by Hubble's Law.

That's how FRW coordinates work -- the reference frames themselves are considered to be receding away from each other at the Hubble velocity HD, regardless of whether a particular reference frame contains an observer. Many people interpret that to mean that in FRW coordinates empty space itself is expanding and dragging the galaxies apart (like a line of dots marked on the surface of an expanding balloon), which is fine if one applies the analogy carefully, but that interpretation is not necessary to describe and calculate the observed redshift.

OK so it is FRW not Hubbles law that imposes these requirements.

 

[nutgeb]

OK it seems that you are not talking about potential because as far as I know the term has not changed meaning since Newton.

As I said, when I responded to your initial question I was referring to the absolute value of the potential, which I could have been clearer about. So apparently we have no remaining disagreement on this point.

Well my knowledge of GR is limited to say the least but from what is being said in this forum aside from geodesic inertial effects on a photon there is a decrease in actual speed with decreasing Schwarzschild radius. This may have negligable local effect but if true would seem to mean significant effect between the intergalactic space and the densely populated galactic space and then the less dense but shorter radius of the solar system and earth.

I was trying to focus my comments on FRW coordinates, which are assumed to have a fully homogeneous matter distribution. I also pointed out that the grativational effect of a single massive body (planet or galaxy) could be modeled separately in Schwarzschild coordinates and then combined with the FRW cosmological redshift. And finally I pointed out that the bit of blueshift resulting from the Schwarzschild mass at the observer would generally be an insignificant component of the total cosmological redshift.

Having said that, I interpret that you are suggesting or asking whether, when applying the Schwarzschild solution in isolation, the proper velocity of a photon decreases as it approaches the central mass, as calculated in the emitter's reference frame. I haven't done calculations on that point, but I believe that while a photon's Schwarzschild coordinate velocity will change, its proper velocity always remains c as calculated in the reference frame of a distant emitter.

In FRW proper distance coordinates, the time coordinate and the radial spatial coordinate remain linear, due to the homogeneous mass distribution, and the photon's proper velocity increases in the emitter's frame, due to Hubble's Law. Conversely in Schwarzschild coordinates, in the distant observer's reference frame the time coordinate slows down (due to gravitational time dilation) and the radial spatial coordinate r expands (due to spatial curvature per Flamm's parabaloid) as the photon approaches the central mass. But I believe the photon's proper velocity remains constant in that reference frame, because proper velocity (change in proper distance divided by change in proper time) is an invariant quantity in Schwarzschild coordinates (with the possible exception of inside a black hole event horizon). But if anyone has additional insight on that subject I'd appreciate some input.

OK so it is FRW not Hubbles law that imposes these requirements.

Hubble's Law exists only in FRW coordinates, and it is a mandatory characteristic of FRW coordinates, so for this purpose the two are one and the same.

 

[nutgeb]

Conversely in Schwarzschild coordinates, in the distant observer's reference frame the time coordinate slows down (due to gravitational time dilation) and the radial spatial coordinate r expands (due to spatial curvature per Flamm's parabaloid) as the photon approaches the central mass. But I believe the photon's proper velocity remains constant in that reference frame, because proper velocity (change in proper distance divided by change in proper time) is an invariant quantity in Schwarzschild coordinates (with the possible exception of inside a black hole event horizon).

I need to clarify this statement.

In terms of the coordinates of a distant Schwarzschild observer, as a photon approaches the Schwarzschild mass the time coordinate t speeds up (relative to proper time) and the radial spatial coordinate r shrinks (relative to proper distance).

The distant Schwarzschild observer might say that proper time has slowed down and proper radial distance has increased relative to Schwarzschild coordinates, but that really is turning the description on its head, since proper time and proper distance are invariants.

Also, while it is true that a photon's proper radial velocity remains constant, i.e. is invariant, as calculated by a distant observer using the Schwarzschild metric, the metric "blows up" as the event horizon is approached, so the Schwarzschild metric cannot be used to calculate a photon's proper radial velocity at or inside the event horizon.

However the variant Painleve-Gullstrand coordinates can be used inside the event horizon, and they portray the photon's radial proper velocity as exactly c at the event horizon and as increasing above c without limit inside the event horizon as the singularity is approached. In this sense the photon's proper velocity inside a black hole event horizon behaves in a way that is conceptually analagous to how it behaves in FRW coordinates beyond the Hubble radius (where the recession velocity HD exceeds c).

 

[Austin0]

I need to clarify this statement.

In terms of the coordinates of a distant Schwarzschild observer, as a photon approaches the Schwarzschild mass the time coordinate t speeds up (relative to proper time) and the radial spatial coordinate r shrinks (relative to proper distance).

The distant Schwarzschild observer might say that proper time has slowed down and proper radial distance has increased relative to Schwarzschild coordinates, but that really is turning the description on its head, since proper time and proper distance are invariants.

Also, while it is true that a photon's proper radial velocity remains constant, i.e. is invariant, as calculated by a distant observer using the Schwarzschild metric, the metric "blows up" as the event horizon is approached, so the Schwarzschild metric cannot be used to calculate a photon's proper radial velocity at or inside the event horizon.

However the variant Painleve-Gullstrand coordinates can be used inside the event horizon, and they portray the photon's radial proper velocity as exactly c at the event horizon and as increasing above c without limit inside the event horizon as the singularity is approached. In this sense the photon's proper velocity inside a black hole event horizon behaves in a way that is conceptually analagous to how it behaves in FRW coordinates beyond the Hubble radius (where the recession velocity HD exceeds c).

Hi nutgeb I think we can limit this discussion to outside the event horizon, yes??

As you have outlined here the coordinate time and distance as measured by Schwarzschild observers is subject to the length contraction and time dilation varying with their radial distance. This implies that coordinate speed of measured photons should increase with decreasing radius, yes?
But we know that local measurements will still produce the invariant c.
This implies that the actual speed of the photon must actually be decreasing or this would not be possible , no?
As I said , this is not my idea but the understanding I have gained from others in the forum who are qualified.

Based on this premise the effect would be most significant approaching the local mass of reception but by no means limited to this.
In this context, scale is not relevant. Approaching a galaxy from free space is entering the zone of a huge massive concentration and it would be expected that the spacetime curvature and resulting effects from this would be significant relative to the flat spacetime of empty space , yes?

In this paradigm the photon would speed up leaving the local spacetime of emission reach a maximum in empty space and then slow down reciprocally as it approaches absorbtion.
At all points still maintaining a measured invariant c because all measurement instrumentation would be equivalently effected by local conditions.
This at least is what I am presenting as a possible relevant consideration in this question.

 

[nutgeb]

As you have outlined here the coordinate time and distance as measured by Schwarzschild observers is subject to the length contraction and time dilation varying with their radial distance. This implies that coordinate speed of measured photons should increase with decreasing radius, yes?

No, as explained in my most recent post, it is the opposite of your description. To a distant observer, the Schwarzschild coordinate speed of a photon slows down as it approaches the central mass. In the case of a black hole, the photon's coordinate speed goes to zero as the photon approaches the event horizon, so it never quite reaches it. But this is just a coordinate glitch, not a 'real' effect.

But we know that local measurements will still produce the invariant c.

Yes, the photon's proper speed will remain invariant at c, measured as it passes by each of a series of stationary observers arranged along the photon's radial path toward the center.

This implies that the actual speed of the photon must actually be decreasing or this would not be possible , no?

Your term 'actual speed' doesn't have a defined meaning. I referred to the photon's 'proper velocity.' Actually my terminology requires more elaboration. In theory the proper speed of a photon is infinite, if it is measured by a clock carried by the photon, which always ticks off zero time. As I explained in my OP, the speed I am referring to is the photon's speed c in the local frame it is passing through, plus the proper velocity of that local frame relative to a distant observer. So I was referring to the photon's velocity in proper distance and proper time coordinates (FRW coordinates), or for short, I'll call it the photon's "proper coordinates velocity."

In the Schwarzschild scenario, the photon's locally measured speed is invariant. A series of stationary observers along the photon's path each will measure its local speed as c.

But as I explained in my last post, the photon's speed in 'distant' Schwarzschild observer t and r coordinates will slow down as it approaches the center, not speed up.

The 'distant' Schwarzschild observer calculates a different speed because its time coordinate is based on a clock that runs faster than those of the local observers nearer the center, and because its local space is spatially flat while theirs is curved.

I don't see any useful way to import the concept of "proper coordinates velocity" from FRW, because every Schwarzschild observer figures the local reference frame of every other Schwarzschild observer who is stationary with respect to the central mass to have zero proper recession velocity. The closest analogy is to use the photon's locally measured velocity, which is always c.

No matter how you look at it in the Schwarzschild scenario, outside the event horizon an inward plunging photon's velocity does not increase and never exceeds c.

 

[Austin0]

As I said , this is not my idea but the understanding I have gained from others in the forum who are qualified.

As you have outlined here the coordinate time and distance as measured by Schwarzschild observers is subject to the length contraction and time dilation varying with their radial distance. This implies that coordinate speed of measured photons should increase with decreasing radius, yes?

No, as explained in my most recent post, it is the opposite of your description. To a distant observer, the Schwarzschild coordinate speed of a photon slows down as it approaches the central mass. In the case of a black hole, the photon's coordinate speed goes to zero as the photon approaches the event horizon, so it never quite reaches it. But this is just a coordinate glitch, not a 'real' effect.

I was referring to local Schwarzschild observers not distant.

But we know that local measurements will still produce the invariant c.

Yes, the photon's proper speed will remain invariant at c, measured as it passes by each of a series of stationary observers arranged along the photon's radial path toward the center.

This implies that the actual speed of the photon must actually be decreasing or this would not be possible , no?

Your term 'actual speed' doesn't have a defined meaning. I referred to the photon's 'proper velocity.' Actually my terminology requires more elaboration. In theory the proper speed of a photon is infinite, if it is measured by a clock carried by the photon, which always ticks off zero time. As I explained in my OP, the speed I am referring to is the photon's speed c in the local frame it is passing through, plus the proper velocity of that local frame relative to a distant observer. So I was referring to the photon's velocity in proper distance and proper time coordinates (FRW coordinates), or for short, I'll call it the photon's "proper coordinates velocity."

How is this? The measurement of an inertial , free falling, particle's velocity by stationary observers along the path increases, yes??
If we consider that particle as following a geodesic in inertial motion then the increase in coordinate velocity is attributed to the contraction and dilation effecting the local measurements.

The local conditions themselves effectuate an inherent icrease in measured velocity.

If a photon is measured by the instrumentation under these same local conditions and is still seen to have a constant velocity then this logically implies that it must be, actually, slower or else the same clocks and rulers would measure an increase,,?

But as I explained in my last post, the photon's speed in 'distant' Schwarzschild observer t and r coordinates will slow down as it approaches the center, not speed up.

The 'distant' Schwarzschild observer calculates a different speed because its time coordinate is based on a clock that runs faster than those of the local observers nearer the center, and because its local space is spatially flat while theirs is curved.

I don't see any useful way to import the concept of "proper coordinates velocity" from FRW, because every Schwarzschild observer figures the local reference frame of every other Schwarzschild observer who is stationary with respect to the central mass to have zero proper recession velocity. The closest analogy is to use the photon's locally measured velocity, which is always c.

No matter how you look at it in the Schwarzschild scenario, outside the event horizon an inward plunging photon's velocity does not increase and never exceeds c.

Unless I am misunderstanding you , this whole premise is based on a concept of increased relative speed of light even if it is still measured locally as c? Isn't that in a way suggesting an actual speed that is different from local measurements?