LHS1 said:
I spent some time to study general relativity recently and I found something that confuse me. According to my unserstanding about general relativity, the coordinate velocity of light should be (1-2GM/c2r)c, while the wavelength should be (1-2GM/c2r)1/2times its wavelength at infinity, for a photon falling from infinity to the surface of a non-rotating star with radius r with an observer kept stationary on the surface, therefore I conclude that its frequency should be (1-2GM/c2r)1/2times its frequency at infinity, that is , its frequency should be decreased.
You should be aware of two types of measurements, those made by an observer that remains at infinity that I shall call "coordinate measurements " and those made local observers that I will "local measurements".
Coordinate measurements.
The coordinate speed of light c' = c(1-2GM/c
2r)
The coordinate wavelength of light w' = w(1-2GM/c
2r)
The coordinate frequency of light f' = f
Note that the basic wave relationship c' = fw' is maintained but c' is not constant. In coordinate measurements it is the frequency that is constant and the energy of the photon does not change as it falls (or rises).
Local measurements.
Imagine a series of local observers maintaining stationary positions at various heights making measurements of falling photons as they pass. According to the observer at infinity the local observers are using clocks and rulers that are slower and shorter than those at infinity by a factor of (1-2GM/c
2r)
1/2.
The local speed of light c' = c
The local wavelength of light w' = w(1-2GM/c
2r)
1/2
The local frequency of light f' = f(1-2GM/c
2r)
-1/2
Note again that the basic wave relationship c=f'’w' is still maintained but this time it is c that is constant.
Local measurements seem to be the preferred measurements of most of the people on this forum, because the speed of light is constant, but there are a number of problems with local measurements. The first is that the frequency of a rising photon gets less and this prompts the question "Where has the energy of the photon gone?". It is not at all clear if the gravitational field in GR is like that of classical physics and absorbs the energy as potential gravitational energy. Some people conclude as a result that GR does not conserve energy. A bigger problem is that distance between two fixed points depends on the direction that the measurement is made in local measurements. For example imagine there are two observers, A and B at fixed heights in a gravitational field. A is low down in the field (say on the surface of a neutron star while B is high up in a tall tower. A sends a light signal up to a mirror by B which is reflected back to A. Observer A times the round trip of the photon and measures the time to t(ABA). B does a similar measurement and sends a signal down to a mirror by A and times the round trip of the photon as t(BAB). When they compare measurements they find the round trip time obtained by A is less than the round trip time obtained by B. If they continue to cling to the concept that the speed of light is constant everywhere they are forced to conclude that the distance (ABA) is different to distance (BAB). This does not make sense when you consider that the outward leg (AB) of A's measurement is identical to the return leg (AB) of B's measurement and the return leg of A's measurement (BA) is identical to the outward leg (BA) of B's measurement. Something isn't adding up. They can carry out another experiment to find out what is going on. They agree to send signals at one second intervals to each other where the intervals are timed by their own clocks. When they do this they see their clocks are clearly running at different rates and A's clock is running slower than B's clock. If they apply logic, they will realize that if their clocks are running at different rates and if they both measure the speed of light to be constant according to their own clocks, then the speed of light high up by A can not really be the same as the speed of light down by B. Now Einstein made great progress by assuming a consistent method of synchronising clocks in Special Relativity and we can make progress here by synchronising the clocks of the two observers in the gravitational field by a slightly different method. What we do is pass the output of the atomic clock through a digital processor and artificially speed up the lower clock or slow down the higher clock (it does not matter which) so that when they send signals to each other at one second intervals they both see the signals arriving at intervals of one second. Now when they measure the frequency of falling light they will see the frequency is not changing and energy is conserved. They will also find that the distance ABA is now the the same as the distance BAB and the height of the tower is the same whichever end you measure it from, using a two way light signal. If they have a series of observers at different heights and compare all their measurements with each other and compare the distance measured by two way light signals with distances measured by physical rulers they will soon work out that all the clocks are running progressively slower the lower down they are, relative to a clock at infinity. If they "synchronise" their clocks by artificially speeding up the lower clocks they will find that radial distances measured by two way light signals agree with distances calculated from measuring orbital circumferences and that space is Euclidean. In other words, the radial distances and circumferences agree with 2 \pi R.
LHS1 said:
However I found a textbook , that I do not fully trusted, on general relativity in introductory level said that the frequency should be (1-2GM/c2r)-1/2times its frequency at infinity. That is , the frequency increases. Should the frequency increase or decrease? If I am wrong, please give me a detailed explanation. Thank you.
The textbook is correct if it is talking about local measurements which it almost certainly is. You must be careful not to mix up local measurements with coordinate measurements and that seems to be what you have done in the first part of your post.
There is also potential for confusion between the terms "local measurement" and "proper measurement". If a stationary observer in a gravitational field measures the length of falling object as it passes that is a local measurement. If the observer is falling with the object then his measurement of the falling object's length is a proper measurement. Proper time of a falling object is the time measured by a clock attached to the object and in the case of photon where we can not physically attach a clock to the photon the proper time of the photon is taken to be zero. The time interval measured by a non inertial stationary observer in the gravitational field for a falling object to fall a short distance as it passes him is a local measurement and not a proper measurement.
P.S. I am not so good at formal definitions so I am happy for anyone to correct my definitions. I defined coordinate measurements as those made by an observer at infinity which is the usual default unless otherwise specified. However with a slight modification of the equations the term "coordinate measurement" can be more generally applied to any observer at a fixed altitude that makes measurements that are not necessarily local. In this case an observer at some arbitary altitude will see the coordinate velocity of a falling photon below him as less than c and the coordinate velocity of a photon above him and falling towards him as greater than c.
I will stay with the constant frequency interpretation which I prefer for a number of reasons, that I hope I made clear in post #3.