23 Gravitational Redshift 2Sep18
Introduction Earlier essays in this series of blogs provided a plain English synopsis of the foundations of General Relativity, presented some heretical comments about Einstein’s Equivalence Principle and introduced the Clock Postulate. In this essay I would like to take a closer look at gravitational redshifts. [Note: In this essay I will again be using references from the heavyweight textbook on gravity by Charles Misner, Kip Thorne and John Wheeler (MTW): “Gravitation” C W Misner, K S Thorne, J A Wheeler Freeman Press, 1970 ISBN 0-7167-0344]
Effects of Gravity on the Propagation of Light The effects of gravity on light are not straightforward. Einstein’s ideas evolved over more than a decade and involved discussions with and by other great scientists such as Max Planck, Hermann Minkowski, Max Born, Willem de Sitter, Max von Laue, Hermann Weyl and John L Synge. Einstein wrote on the subject repeatedly, e.g. in 1907, 1908, 1911, 1915 and 1916. His 1915 paper contained significant corrections to his 1911 formulae. His 1915 work opened with the remark “‘I return to this theme because my previous presentation does not satisfy me.”
Einstein came to the conclusion that one of the basic tenets of the Special Theory of Relativity - the constancy of the velocity of light – had to be abandoned when gravity is taken into account. Einstein embraced this consequence and made it the basis of a further prediction – the bending of light passing near a massive body. His 1911 prediction for the light bending was the same as that of classical physicists.
In 1912 Einstein’s approach to modeling the effects of gravity on light using only his principle of equivalence ran into problem after problem. This motivated him to begin studying in earnest the differential calculus of Ricci and Levi-Civita as applied to curved geometrical manifolds (involving tensors, co-variant derivatives, Christoffel symbols and the like.)
Einstein revised his 1911 calculation in his 1915 paper and came up with a prediction twice as large as the original estimate. Sir Arthur Eddington famously claimed to have verified this prediction several years later in 1919. [In spite of World War I, Eddington had received a copy of Einstein’s work on General Relativity and he quickly became an early supporter of its ideas. Eddington came up with the idea of measuring the bending of light during a total eclipse and he obtained support from the Royal Astronomical Society to do so. I think this is a nice example of scientific cooperation transcending national hostilities.]
Einstein’s theory of General Relativity eventually contained three effects of gravity on light: 1. Gravity slows the speed of light, 2. photons climbing out of a gravity well arrive redshifted, and 3. gravity bends the path of light.
The degree to which the second and third effects are were predicted by Einstein’s 2015 theory became important tests for the new theory, along with the calculation of the anomalous precession in the perihelion of Mercury (see latter essays).
As to the redshift, Einstein wrote in a letter to Arnold Sommerfeld in 1912, ”The clock goes more slowly if set up in the neighborhood of ponderable masses. From this it follows that the spectral lines of light reaching us from the surface of large stars must appear displaced towards the red end of the spectrum”.
This makes it clear that Einstein was attributing some or all of the gravitational redshift to the time dilation caused by gravity, which in turn is intimately connected to the speed of light in gravity (for as we discussed earlier, the very meaning of time is intrinsically intertwined with the concepts of distance and the speed of light).
I like these comments by two American writers in 1980 (John Earman and A Glymour): “Einstein’s early derivations of the red shift show his most characteristic style of work - heuristic, allusive, sometimes baffling, but unfailingly fruitful.” They go on to say “Altogether, there may be no other single topic which so vividly illustrates the intellectual ferment, the styles of work, the profundity and the confusion associated with the general theory of relativity.”
We can argue at length about the exact meanings of the language used by Einstein and others to describe the three effects, as many good physicists, mathematicians and philosophers have done for the last century, especially if they are inclined to the modern view that gravity is an illusion created by spacetime curvature.
The more important thing is that the three effects led to new predictions for the size of gravitational redshift and gravitational light bending which became early tests for the new Theory of General Relativity.
The success of Einstein’s new general theory in predicting the size of gravitational redshift and light bending effects has led textbook writers to assert that classical physics has shortcomings that required the genius of Einstein’s curved spacetime theory to correct.
However, I think that the experimental results are what Sir Isaac Newton, Pierre-Simon Laplace and many other great classical physicists scientists over the previous three hundred years had already anticipated to some degree and would not have been surprised at all to see confirmed.
I think that General Relativity is a powerful new approach that brings in a whole new class of mathematical tools and so lends itself to a better description of some small effects in extreme situations. But I also think that if you add Special Relativity and the fact that gravity slows the speed of light to classical physics you can get the same answers. Certainly for the three effects on light (the first result is axiomatic) and possibly also for the anomalous precession in the perihelion of Mercury.
This is what the next few essays are going to examine. Starting with gravitational redshifts.
Gravitational Redshift - The Long Hollow Rocket Imagine that the very tall elevator shaft in the previous essay has become a long hollow rocket which is in deep space somewhere and that this rocket is accelerating at a high constant rate forwards. Imagine there is a source of photons with a very tightly defined frequency range (a laser for example) situated at the back of the rocket and that this has just fired a burst of photons towards the front of the rocket. When this burst of photons was sent on its way, the back of the rocket was travelling at a certain speed. Due to the rocket’s acceleration, by the time the photons reach the front of the rocket the front of the rocket will have reached a higher speed. In other words there will be a relative speed difference increase from the back to the front of the rocket due to the acceleration that takes place while the photons are in flight.
Detectors at the front of the rocket will find the arriving photons to have lower energy (hence lower frequency and longer (redder) wavelengths) than they had when the photons started. Furthermore, if the arrangement of laser and detectors is reversed, then the detectors when positioned at the back of the rocket will find photons arriving from the laser at the front of the rocket to have acquired extra energy and thus been blue-shifted. It is a straightforward Doppler effect. (There will be infinitesimal Lorentzian effects as well but these can be safely ignored for our purposes).
Einstein’s Equivalence Principle says that the above situation is the same if the rocket is actually a very tall elevator shaft sitting in a uniform gravitational field. Photons fired upwards in the shaft will arrive redshifted and photons fired downwards in the shaft will arrive with a degree of blue shift.
The redshift effect has been confirmed by experiments such as that of Pound and Rebka at Harvard in 1960.
It is possible to persuade ourselves that light must be red shifted in this way using Einstein’s discovery that energy and mass are equivalent to each other, and applying this to a thought experiment (see MTW p187) as follows.
Imagine that a well defined amount of mass falls through gravity and does some work on the way (turning a treadmill for example). It is then entirely converted into photons that are beamed back up to the starting point. Unless these photons lose some energy they could be turned back into the same initial starting mass and the process could be repeated endlessly, performing work on every loop. But this would violate the principle of Conservation of Energy. Hence Einstein reasoned that the photons must lose energy on their way back up to the starting point.
Does Gravitational Redshift Imply Spacetime Curvature? (MTW p 187) “An argument by Schild (1960, 1962, 1967) yields an important conclusion: the existence of gravitational redshift shows that a consistent theory of gravity cannot be constructed within the framework of special relativity”.
(MTW p189) “Schild’s redshift argument … does say … quite unambiguously, that the flat spacetime of special relativity is inadequate to describe the situation, and it should therefore motivate one to undertake the mathematical analysis of curvature.”
The Schild argument builds on the experimental demonstration of gravitational redshift by Robert Pound and Glen Rebka at Harvard University in 1960.
In 1958 a way had been found to use the Mossbauer resonance effect to emit and absorb gamma rays in a very narrow and precise frequency range using solid samples containing radioactive Fe57. Pound and Rebka made use of this discovery and placed two such samples vertically in a tower at Harvard with a height difference h (and so at a gravitational potential difference of gh in the language of Newton).
General Relativity predicts photons emitted from one sample will no longer be absorbed by the other. But if the absorber is vibrated so that it obtains a range of vertical motions relative to the source, the resulting Doppler effects can restore some absorption.
It is common to see this experimental result described mathematically as ∆t2 = (1 –(Φ2 – Φ1)/c2) ∆t1 where (Φ2 – Φ1) = gh is the difference in gravitational potential.
In essence Schild invites the reader to consider a Lorentz reference frame aligned to the Earth and containing an electromagnetic wave generator at one level and a suitable detector at a higher level, both at rest with respect to each other. Schild’s argument goes like this: The bottom generator emits a wave of exactly N cycles of well defined frequency √ in time interval T and this is received by the top detector. The observer at the top detector is asked to determine how long this signal lasts. In flat spacetime the answer should be T, since the top and bottom observers are at rest with respect to each other. However, the signal undergoes a gravitational wavelength change, lengthening as it climbs up towards the top observer. N cycles of a longer wavelength should last longer than T. The conflict can only be resolved if spacetime is curved.
I agree that gravitational redshift occurs in reality, and I also accept that gravitational time dilation occurs in reality. And if the time dimension is significantly slowed by the presence of gravity, then the usual Lorentz-Minkowski flat four dimensional spacetime framework becomes suboptimal for describing what is going on in the physics.
However, I do not accept the argument that the Pound-Rebka demonstration of gravitational redshift proves that it is necessary to invoke curvature in all four dimensions of spacetime because I think the Schild argument is inherently flawed and in any case it would only introduce a degree of flexibility in the time dimension.
For a start, the Pound-Rebka experiment example used by Schild does not take place in a Lorentz reference frame at all. Very few experiments do. Inertial setups are so rare and to be almost non-existent. Orbiting space stations come close but even then there is still rotation relative to the so-called “fixed stars”. But my concern is mainly about the misuse of wave concepts.
What is the ‘wave’ talked about by Schild (as described by MTW)? It seems Schild is thinking about the emissions being electromagnetic waves with spatial properties related to their wavelength and temporal properties related to their frequency. He talks as if the wavelength gets ‘stretched’ in transit between the bottom and the top of the tower. He talks as if the signal has a well defined frequency in time T and hence acts as a type of clock.
But Einstein himself was instrumental in demonstrating that electromagnetic radiation takes the form of photons. These are emitted with precise energy level. If they have to climb against a gravitational potential then when they arrive they are detected as having less energy. That is the relatively simple experimental fact.
And as I discussed earlier (and summarise in the box below) I think the whole schizophrenic particle/wave duality concept of light is seriously old fashioned and that it can be resolved simply by following the evidence with a fresh and open mind.
I think of the precise electro-magnetic emissions in the Pound- Rebka experiment as being phots. They have no length, they do not wriggle about as they travel and they are not little clocks. They are just packets of energy with some intrinsic properties that are only revealed upon absorption.
Schild is suggesting that the bottom generator emits a wave that has a well defined frequency in time T and therefore acts as some sort of clock. Now it is perfectly possible to incorporate a beat into an overall emission of lots of phots. Just synchronise their phase at time of emission and change this in a structured manner as time progresses. It is what happens all the time in a radio signal. The signal is encoded in the overall pattern. But not in each individual phot.
Do not get mixed up between what happens to the pattern of phots and what happens to the phots themselves, which is what I think Schild (and others) have done. And forget about anything with a finite length being “stretched” somehow. Phots have no length.
All the Pound Rebka experiment does is demonstrate that phots emitted in precise circumstances at one point in a gravitational field and absorbed at another point in the gravitational field lose energy consistent with the change in gravitational potential. This shows up upon their absorption/detection/destruction as a decrease in the frequency of their embedded signal. If you insist on using the phots as a way of standardising time in reference frame covering the whole experiment, you are entitled to conclude that time runs more slowly at the bottom of the tower than at the top.
So does a gravitational redshift as demonstrated by a single Pound and Rebka experiment demonstrate that gravity has to be modeled by a fully curved spacetime approach? I do not think so. In fact I think that a gravitational redshift occurring between two specific points in flat spacetime can be understood perfectly well just using classical physics augmented by Special Relativity and the recognition that gravity slows the speed of light (and hence time).
A single gravitational redshift experiment in one particular location is not proof of full spacetime curvature. However, if you consider how to interpret the results of many different redshift experiments spread around a gravitationally perturbed region of the Universe at more or less the same time, then the argument becomes stronger.
Take Earth for instance. If we consider a whole set of Pound-Rebka experiments occurring at different locations around the Earth, then while each experiment might have something to say about its local spacetime environment and local inertial reference frames, the only way to connect all the frames is to admit curvature into the dimensions of spacetime more generally. This is the conventional route to Einstein’s General Relativity.
So I’m saying that while curving all the dimensions of spacetime is not necessary to understand gravity per se, it is still a very useful approach to modeling some subtle effects in physics in gravity fields surrounding massive celestial objects.
How to Interpret Gravitational Redshifting? Pound-Rebka and many others showed that gravitational redshift does occur, and a variety of thought experiments suggest that this is a perfectly reasonable outcome. But how should we interpret the results? It is a debate that went on for decades from about 1911 onwards and it is a question that is still open, although many minds are not.
I will let the photons speak for themselves. Let us turn them into little cartoon characters. On arrival at the top detector the photons could explain what has happened in one of three possible ways: (1) “Hi, we are from bottomland. Time runs slow down there, so please excuse us if we are a bit slow. We are all slow down there. Apart from that we are exactly like you.” (2) “Hi, we are your identical counterparts from bottomland. We have had a tough journey and now we find that we have to give up some of our energy in the form of a tax. So please excuse us for being a bit redder than when we started out.” (3) “Hi, we are your identical counterparts from bottomland. You guys seem to have been accelerating upwards while we were travelling. We can’t climb aboard your detector, even though it is identical to the one that gave us birth, unless you lower the bar a bit and retune it to a lower frequency, or push it towards us to shake off all that extra speed you’ve acquired.”
The first explanation suggests gravity causes time to slow down and any and all processes that involve time to slow down as well. This by itself is enough to distort the time dimension in a four dimensional spacetime reference frame. But it does not say anything about curvature in the three spatial dimensions and hence is not an argument for Einstein’s curved spacetime ‘geometric’ model per se.
The second explanation is akin to a standard Newtonian approach. Newton thought of light as consisting of “corpuscules” and fully expected them to be able to be influenced by gravity. See the following essay about the bending of light by the Sun.
The third explanation of gravitational redshifting uses the Einstein Equivalence Principle to suggest that what is going on is that both the bottom and the top of the tower are being accelerated in curved spacetime. This is what causes the redshift as per our discussion of the long hollow rocket. You could say that this is an explanation in terms of the Doppler effect.
A student of Special Relativity might not be surprised by gravitational redshifting since, if a photon is energy, and energy is equivalent to mass, and mass loses energy when it climbs out of a gravity well, then why would anyone not expect a photon to lose energy also? So such a student might be inclined towards explanation (2).
One of things that intriques me about gravitational redshifting is this. If the photons arrive at the top of the experiment with less energy than they started out with – where did that energy go? If the photon were a little rocket then it would end up in the heat and kinetic energy of the exhaust gases. If the photon was a solid projectile the lost energy is apparent in the gradual loss of kinetic energy. If the photon was pulled upwards by a string it would clearly come from whatever was winding up the string. But in the case of a photon it starts off with one amount of energy and arrives with an amount that is lower than that of photons being created in exactly the same way but at the ‘higher altitude’. Where did the difference go?
I think that the question goes to the heart of understanding gravity and hence is a quite profound. But most lecturers will just say “It has ended up as a reduction in negative potential energy” and leave it at that. Personally I think that this ‘papers over’ a gap in a better understanding of the situation. The same thing happens is you ask questions like – what gives matter its mass? or what gives mass its inertia? or why does a moving object want to travel in a straight line? Just giving physical phenomena names instead of explanations tends to block our minds to deeper understandings.
The first and third of the explanations require a non-flat spacetime reference frame due to distortion in the time duration dimension.
If one of the answers is correct it does not necessarily mean that the others are incorrect. In principle the correct answer might depend upon what point of view you are using. Then the best answer is then the one that is consistent with the point of view you are using. And the best point of view is generally the one that is the most convenient for your purposes at hand.
Furthermore, it is also possible in principle that the best answer requires a combination of the various explanations. In fact I think it does, as I will explain later.
So what about explanation (3) that interprets the redshift as a Doppler effect? My take on this is that if you want to adopt the Einstein Equivalence Principle, and if you want to replace the greatest force in the Universe with the mathematical trickery of curved spacetime, then this is the explanation you should use. It also gives the right answer, so it becomes a matter of choice.
But the fact that you could prefer Explanation (2) shows that the curved spacetime approach is not the only way to understand the Pound Rebka experiment. And even if you do want to use Explanation (3) you should note that this does not require you to use Einstein’s full model. If you do use Einstein’s General Relativity model then it is only the perturbation of the time term that is needed in order to come up with the observed results for gravitational redshifts. The full field equations are not needed and there is no need to call upon any warping in the spatial aspects of the spacetime geometry.
Gravitational Redshift in the Solar Spectrum Light reaching Earth from the Sun’s surface has climbed a long distance up the Sun’s considerable gravity well, and has fallen into the Earth’s smaller gravity well. The light itself comes from a large number of sources with known spectral frequencies, but these spectral lines are complicated by the extreme thermal motion of the sources, the Sun’s rotation and the Earth’s own motion. All of this creates a blur of Doppler shifts. Nevertheless it is possible to screen and correct for the blurring effects and the resultant redshifts are consistent with the expected result.
Early attempts to measure the gravitational redshift of light reaching Earth from the Sun were plagued by practical difficulties. The earliest results tended to disprove Einstein’s predictions. When Einstein’s fame and reputation soared towards the end of the second decade of the 20th century, the trend reversed and it became more fashionable to produce results corresponding to Einstein’s predictions. Modern results do confirm Einstein’s predictions.
It is interesting that the earlier attempts to explain the spectral shifts in light from the Sun did not use General Relativity. This shows that while on one level Einstein’s general theory won wide acceptance, on another level there was a reluctance to fully adopt the curved spacetime approach. General Relativity was thought of as being impressive and interesting, but also a bit too weird and not to be taken literally.
The Best Way to Interpret Gravitational Redshifts? General Relativity has now become the orthodox paradigm but even today arguments continue about how best way to tie in the experimental evidence about gravitational redshifts. Some authors/teachers prefer one type of explanation, others prefer another.
I’m pretty sure many students find this confusing. But I also think that there may be more than one way to look at it. I do not think that one view corresponds to “reality” and the others are fallacies. They are all just mental models created for our own convenience of understanding.
Here is an analogy. A cone can be seen as a triangle from one perspective and a circle from another. The real nature of a cone transcends both views.
In this spirit I object to those who insist that spacetime curvature is real and gravity is not real. I say that you can use a curved spacetime model if you like, but it is only a model. Likewise you can hold the view that gravity is a real force of nature, but you still also have to recognize the lessons revealed to us by Einstein.
My preferred way of explaining gravitational redshift is as follows. When a photon reaches a zone of space where everything has a higher gravitational potential than things did where the photon came from, it arrives in a new environment. It may have been created in outer shell of a certain type of atom, but it now finds itself unable to join similar situations in similar atoms in the new environment. It has to pay a tax to be allowed to join in. Its energy wallet now longer buys what it used to. The photon can now longer afford the sort of home it came from. It has to settle for a lower energy type of accommodation. One with lower energy levels. For example, a photon than came from a green home might have to settle for a new home in the red light district.
So if we go back a couple of sections and eavesdrop on our animated photon’s conversation when they arrive, what they are saying is (2) “Hi, we are your identical counterparts from bottomland. We have had a tough journey and now we find that we have to give up some of our energy in the form of a tax. So please excuse us for being a bit redder than when we started out.” And their newfound friends say “Don’t worry about it. It happens to everyone. And you haven’t lost any value – it is just that some of your energy is now embedded into your relationship with your new environment all around. You can have it back again should you return home.”
Conclusion Gravitational redshifts can be described in Einstein’s General Relativity model but it is not necessary to invoke the full field equations and curvature in all of the dimensions in order to do so. The basics effect was predicted well before Einstein using nothing more than Newtonian gravity and Galileo’s Weak Equivalence Principle.
Once Special Relativity is taken into account the phenomenon can still be understood in terms of differences in gravitational potential.
The mystery is why this phenomenon is considered to be one of the proofs that General Relativity is the only valid way to look at gravity. I think that what happens to photons encountering changes in gravitational potential can be described without reference to Einstein’s General Relativity field equations at all.
Here is a bit of basic logic. “If General Relativity is to be a useful model then it must not contradict the evidence of gravitational redshifts.” Let us agree that this is true. Then we also have to accept the contra-positive argument that goes as follows. “If General Relativity contradicts the evidence of gravitational redshifts, then it is not a useful model.” What we do not have to agree is the argument that “If General Relativity is consistent with the evidence of gravitational redshifts, then it is a useful model”. This statement may or may not be true. And we certainly do not have to agree without question the insistence by many modern physicists that because General Relativity is a useful model then its method of approach is the only interpretation of Nature that we should agree to be “reality”.
On this I am pretty sure that Einstein would agree.
