Hi Little Bang and John,
Little Bang wrote:If I do an experiment to test the speed of light here on Earth I get a value for c. If I take all my equipment to the moon and use the same clock that was used on Earth and measure c I will get a slower value for c because my clock is running faster than on Earth?
JohnDuffield wrote:No, you will measure c to be the same.
That's because you will define the second as
“the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom”. This is microwave radiation, and light in the wider sense. To define your second you effectively count microwaves going by, and when you get to 9,192,631,770 you say a second has elapsed. Then you use the second along with light to define the metre as
“the length of the path travelled by light in vacuum during a time interval of 1⁄299,792,458th of a second”. Note that you will use the motion of light to define the second and the metre. And then you will use them to measure the motion of light. That’s why you always measure the speed of light in vacuo to be 299,792,458 m/s. It doesn’t matter how fast the light is moving, you will always say the speed of light is 299,792,458 m/s. See
http://arxiv.org/abs/0705.4507 re the tautology.
John's answer is absolutely correct. However please let me interject for the sake of it's pure curiosity value... there are other odd questions that might be crossing your mind at this time as well. They certainly did in mine.
The really peculiar (subtle) fact about this measuring of the speed of light on the Moon and on the Earth, or anywhere else in the Universe for that matter, while the value of C would experimentally show exactly the same values in all cases everywhere in the Universe, and despite the absolute truth of John's reply, time on Earth is "
actually" elapsing more slowly
RELATIVE to the time elapsing on the Moon.
What I mean by "time" is physical time as experienced by things that age such as ourselves and the lifetimes of unstable atomic particles. It is also a form of "local time"... time experienced at a single locality. And that local time is an effect independent of the manufacture of clocks or of any timepiece necessary for use in your Laboratory... and is an effect that has even been recorded on successive stairs in a single building's stairwell, being that the deeper down the stairwell you go (while still being above the surface of the earth) the stronger gravity is and so the slower time proceeds relative to an identical atomic clock left on a step or two above it. The effect is relative to the local value of "g" or g', the acceleration due to gravity at a place, a property that varies over the surface of the earth slightly
(and also varies within the body of the Earth according to g' = ( r/r_{e} )g vanishing at it's center). Less obviously there are other effects if you move the clock (and your Laboratory) around. It is also dependent on the history, where the "clock", or the collection of instruments, have been in the past and the exact path and velocity they all took to get there. The effect is accentuated by being on the Moon where the surface gravity is only 1/6 that on Earth, and that is an effect is due mostly to Einstein's General Relativity (GR). While on the Moon these clocks are aging faster than equivalent clocks left on the Earth due to this General Relativistic effect plus a smaller contribution due to Special Relativity. To calculate accurately a value for elapsed time, a further correction for the velocity of the Moon around the Earth would also be necessary, due to Special Relativity (SR) but this secondary effect is of a smaller magnitude than the effect due only to gravity. At relatively low velocities, way beneath the actual speed of light, the effects of GR are greater than the effects of SR... because Special Relativity only becomes more important at much higher
relative speeds humans and their clocks ever encounter in their normal lives. Still... the proper motion of a GPS satellites must be separately and individually accommodated for at your GPS (or cell phone) through a process of multiple triangulation, to finally provide a true accurate account of your position.
The clock on the Moon's time intervals, let say nanosecond intervals, would "elapse" slightly faster when compared with the same observed intervals on Earth. Or putting this the other way around observers on Earth would notice the clock on the Moon running ever so slightly faster (mostly due to GR) than identically fabricated clocks on the Earth. Of course these kinds of "accurate" clock would be made to ignore local gravity and not contain any pendulums or other gravity sensitive components to fairly provide this "time". You could correct for this variable "ticking rate" by mechanically (or electronically) adjusting the "rate" of clock on the Moon (or on the Earth) and in the case of our GPS satellites,
that is exactly what they do all the time to maintain about a 20-30 nanosecond per day accuracy overall. There are corrections applied to all the clocks on board all the GPS satellites of an order of about
7 microseconds of "
falling behind" per day relative to accurate clocks placed on the Earth, as a result of
Special Relativity, and
45 Microseconds "speeding ahead" per day, as the result of
General Relativity. So every day the clocks on the satellites are maintained to a mutually agreed window of synchronized values of about 20-30 nanoseconds all up to give the relative precision at the Earth surface needed to determine your position to about 5 meters in accuracy. This artificial way of mutual synchronization between orbiting satellites, uses both the velocity of the clocks in space (SR correction) and also the effect of "lower gravity" experienced in orbit around the Earth (GR correction). Notice the effect due to GR is usually much larger than the SR correction. Also note that they are in opposing directions in low gravity and at low velocity relative to C. Even though satellites are moving at about 14,000 km/hr in various orbits that is "low" relative to 300,000 Km/sec. These signals are used on Earth by your cell phones to determine your position at the Earth surface, sometimes down to as little as a few meters. Special differential instruments can "refine" this distance to overall inaccuracies of less than 1 cm. However without adjusting those clocks, and assuming they were identical in every respect, they will all show a natural phenomenon that causes the clocks to "age" slightly differently relative to each other and their history. In the case of GPS satellites and without correction those initially synchronized clocks would wander around the "fix" by as much as 10 Kilometers per day quite randomly about the surface of the Earth. From these matters we know that there is no such thing as a universal time everywhere, or anywhere, and in regions where gravity and speed are near to the theoretical limits such as with being in proximity to a Black Hole and effects noted in high velocity atomic particles moving near the speed of light, these effects are very considerable indeed.
All these facts about Special and General Relativity can be regressed back to one single Law of the Universe, Einstein's Greatest Discovery was that all acceleration, no matter from what source (gravitational or inertial... or even centripetal where the effect includes "frame dragging" through an application of Mach's Principle), lead to the exact same Special and General Relativistic effect... this is called
Einstein's Equivalence Principle which subsumes all of Einstein's Work in Relativity:
Einstein's happiest thought (1907): "For an observer falling freely from the roof of a house, the gravitational field does not exist". Conversely, an observer in a closed box—such as an elevator or spaceship—cannot tell whether his weight is due to gravity or acceleration.
These two effects can be expressed in the same terms and to this point in time no technical discrepancies with GR or SR are known through experiment, even though this issue has since been tested many times. But you never know, this Principle might fail at different scales in the quantum or galactic realm eventually. Quantum Jumps are one clear example of a "violation"... or not?
A suitable reference for GPS devices and Special and General Relativity is
here.