The affect of spacetime curvatures on the speed of light

[JF131]

I understand that the speed of light can be slowed down when it passes through different mediums.

I also understand that general relativity describes how space time is cured and this curvature depends on the mass of local objects and that it can influence the direction of light.

So my question is, does the curvature of space time have an impact on the speed of light?

 

[WannabeNewton]

Keep in the mind that the idea of the speed of light being reduced upon entering dielectric media with different indices of refraction is a simplistic classical viewpoint. QM gives a more detailed and accurate explanation of what really happens.

As for GR, the local speed of light is always $c$ and space-time curvature does not change that.

 

[Psychosmurf]

Yes, it's called the Shapiro delay.

 

[bahamagreen]

I've wondered about this, too.

When light is traveling a straight line, its speed tangent to the "curve" of the line is the same as its speed in the straight line because the tangent to the line and the line itself are the same line... there is only one vector (or two names for it).

When the light is traveling a curved line, there is the tangent vector, an "inward" vector toward the mass, and the resultant vector that becomes the light's curving path replacing the light's path on the straight line.

If this is correct so far, when the path is curved, which vector (tangent component or resultant) is considered the speed of light?

Since the tangent is shorter than the resultant, I would think the tangent can't be it, it must be the resultant which is the light's direction and c magnitude... but this seems to make the light's tangent vector along the curve have to be <c in order for the longer resultant to be c.

... which might be fine, but looking back at the straight line example, it makes me wonder, since in at least that case it is the tangent that is c, and in a sense the "local" measure on the curve?

 

[Nugatory]

When the light is traveling a curved line, there is the tangent vector, an "inward" vector toward the mass, and the resultant vector that becomes the light's curving path replacing the light's path on the straight line.

It works the same way as a massive object following a curved path; at any given moment the velocity vector is tangent to the curved path. The "inward" (at right angles to the tangent vector) component of the velocity 4-vector is zero; what's not zero is the "inward" component of the acceleration 4-vector, which is the derivative of the four-velocity.

This might be easier to visualize if you think about tying a string to a rock and then swinging the rock in a circle. The speed of the rock as it goes around the circle is constant and the velocity vector is always tangent to the circle. What's perpendicular is the radial force from the string pulling on the rock, so the direction of acceleration is inwards. We end up with the radial component of the velocity, $\frac{dr}{dt}$, equal to zero but the radial acceleration $\frac{d^2r}{dt^2}$ non-zero.

 

[WannabeNewton]

If this is correct so far, when the path is curved, which vector (tangent component or resultant) is considered the speed of light?

The local speed of a light wave as measured by some observer is the magnitude of the projection of the wave 4-vector onto the local simultaneity slice of the observer divided by the projection of the wave 4-vector onto the observer's 4-velocity. This will always be $c$ as is easily checked.

 

[pervect]

I understand that the speed of light can be slowed down when it passes through different mediums.

I also understand that general relativity describes how space time is cured and this curvature depends on the mass of local objects and that it can influence the direction of light.

So my question is, does the curvature of space time have an impact on the speed of light?

This is sort of a trick question, as the answer depends on what you mean by "the speed of light".

If you use local clocks and local rulers, the speed of light is always "c". If you use coordinate clocks and local rulers, or coordinate clocks and coordinate differences (which I try to discourage because it's really wrong to conflate coordinate differences and distances) then the speed of light is not always "c".

I was tempted to go into the difference between coordinate clocks and local or proper clocks, but I decided it would be better to wait to see if there was interest.

Some other points:

The distribution of matter can't be described by assigning a single number (a density) to every point in space-time. You need a more elaborate structure known as the stress-energy tensor, this structure is a rank 2 tensor.

The curvature of a 4-d space-time can't be described by a single number either. It's not a simple number.

You can relate the correct description of the distribution of matter (which is the stress-energy tensor, a rank 2 tensor ) to a correct but compacted description of curvature (the Ricci curvature tensor, another rank 2 tensor) via Einstein's field equations. The relationship then becomes very simple - the Ricci curvature is proportional to the stress energy tensor.

See for example Baez' "The Menaing of Einstien's equation" http://math.ucr.edu/home/baez/einstein/

 

[Naty1]

Hi JF...

does the curvature of space time have an impact on the speed of light?

Depends on just what you mean...as already noted.
LOCALLY
Locally the speed of light in a vacuum is always 'c'. Your local wrist watch always ticks at the same fixed steady pace and so you always measure the speed of light as 'c'. This is like making a measure in flat spacetime, no spacetime curvature. [The Lorentz transform is the relativistically-correct transform from one flat-spacetime reference frame to another when the relative velocities of the two frames is smaller than c.] Everybody, near and far, sees light at 'c' regardless of their own velocity in flat spacetime meaning no gravity. Space contracts and clocks slow among distant observers in a way to maintain constant 'c'

Nugatory nailed one aspect of the LOCAL description:

...at any given moment the velocity vector is tangent to the curved path. The "inward" (at right angles to the tangent vector) component of the velocity 4-vector is zero; what's not zero is the "inward" component of the acceleration 4-vector, which is the derivative of the four-velocity.

DISTANT OBSERVATIONS

The 'view' from a distance can be described by the Shapiro delay in curved spacetime. CURVED spacetime means, general relativity, gravity, where gravitational potential causes distant clocks to run at different relative rates. [It's a coordinate based view.] Each local observer still sees their own clock tick at the same regular pace, but see the other guy's clock as ticking differently from their own!

As pervect notes, such 'coordinate measures' may result in a measured speed different from c in curved spacetimes. [GR makes predictions based on invariants not coordinates; local measures are invarient. Distant measures, are not. That's why you often see things like: "The speed of light is always 'c' as a shorthand, meaning 'locally' because THAT is what GR predicts.]

As an example of the effect of variable elapsed times between observers, a clock at the top of a high building runs ever so sligthly slower than an identical clock at the bottom. So each will measure the other person's lightspeed, the 'distant' light speed, differently than their own local measure. Each thinks their speed is 'c' [locally]. Clocks on GPS ground stations run at different rates from identical clocks orbiting earth...corrections must be made for accurate position calculations.

In such curved spacetime, the Lorentz transform cannot be used to make ‘adjustments’ to bring observers together as in flat spacetime.

It takes a while to put the overall picture together, so be patient and stick with it...

 

[JVNY]

Yes, pervect, I would be interested in more on coordinate clocks etc., and why you discourage the conflation. The Shapiro delay is very interesting and has important consequences, including in communicating with spacecraft . In addition, light travels at less than c in an accelerating reference frame, which is also interesting (and seems to be a good illustration of the equivalence principle). It seems a useful description to say that light really does travel at less than c in these circumstances.

 

[Dale]

Yes, pervect, I would be interested in more on coordinate clocks etc., and why you discourage the conflation.

It is discouraged because you can make coordinate-dependent values assume any value simply by changing your coordinates, and no coordinate chart is preferred over any other. In other words, coordinate-dependent values don't tell you about the physics, they tell you about your coordinate system.

It seems a useful description to say that light really does travel at less than c in these circumstances.

I don't think that the word "really" is ever useful in any circumstance. Just state what is meant: "the coordinate velocity of light is less than c in such a chart". That is complete and informative and gains nothing by the addition of the word "really".

 

[JVNY]

OK, thanks. It would still be great if pervect would explain more about the coordinate v. local clocks (if still available and willing) because coordinate time does seem to be important. For example, Neil Ashby has stated that GPS time is an example of coordinate time, and I have read that the designers of the GPS system had to consider the possible effects of Shapiro delay (but determined experimentally that the delay was negligible).

 

[WannabeNewton]

What exactly do you need explained regarding the *general* difference between coordinate values and values resulting from local measurements made in local Lorentz frames? Would an example help?

 

[Naty1]

JVNY...regarding coordinate time

You'll get considerable insights from pervect's post:

If you use local clocks and local rulers, the speed of light is always "c". If you use coordinate clocks and local rulers, or coordinate clocks and coordinate differences (which I try to discourage because it's really wrong to conflate coordinate differences and distances) then the speed of light is not always "c".

I essentialy provide a 'proper' time DESCRIPTION under LOCALLY in my prior post, and 'coordinate' effects under DISTANT OBSERVATION.

You own local clock, in your frame is your proper time. Most everybody else's clock, say in a different inertial frame or a different gravitational potential,or even local to you but accelerating, will result in coordinate time differences between your clock and theirs. Their local clock in their frame is their proper time.

EDIT: So the result in my prior post is that when you use proper elapsed time, locally, you measure light at speed 'c'; when you calculate the speed of light in a distant environment, you usually get a result other than 'c'...a 'coordinate' result.

 

[pervect]

Yes, pervect, I would be interested in more on coordinate clocks etc., and why you discourage the conflation. The Shapiro delay is very interesting and has important consequences, including in communicating with spacecraft . In addition, light travels at less than c in an accelerating reference frame, which is also interesting (and seems to be a good illustration of the equivalence principle). It seems a useful description to say that light really does travel at less than c in these circumstances.

Coordinate clocks assign a number, which is the coordinate time, to every event. An example of a coordinate clock is TAI time, international atomic time.

Proper clocks measure elapsed time (more on those later).

I'll discuss this extensively, because I believe in specific examples - though the above abstraction, minus the main examples, is the point I'm trying to convey.

http://en.wikipedia.org/wiki/International_Atomic_Time

International Atomic Time (TAI, from the French name Temps atomique international)[1] is a high-precision atomic coordinate[2] time standard based on the notional passage of proper time on Earth's geoid.

TAI time is the sort of time that underlies the usual civil measurement of time, called "Coordinate Univeral Time". The difference is that "Coordinate Universal Time" has leap-seconds added as needed, in order to keep the time at which the sun is directly overhead as noon.

Proper time, can be thought of as the time measurement made by some particular clock, or wristwatch. If you think of a clock as something cyclic (which is usually the case), you just count the number of cycles that occurred between two different events to get the proper time, i.e. how often the clock "ticked". In a typical cesium atomic clock, the "ticking" is counted electronically and you get the proper time just by subtracting the digital readout of the clock at two different events. Note that proper time can and must be measured by a single clock - if you need to synchrornize clocks to make a measurement, you are using some sort of coordinate time, not proper time.


If you study the defintion of TAI time, you'll see something interesting. TAI time matches proper time only at sea level - more formally "on the surface of the geoid". If your atomic clock is above or below the geoid ("sea level"), the proper clock, physically stationary on the Earth's surface, will tick at a different rate than the abstract "coordinate clock".

The abstract coordinate clock is not a physical clock at all, it does not measure the passage of time. It's underlying purpose is to assign an unambiguous time coordinate to every event, while the function of a physical clock is to measure the actual passage of time at a specific location (in the general case, along a specific worldline, which may for instance represent a moving object rather than a stationary one.)

The speed of light is constant when measured by proper clocks. It's not constant anymore after you "adjust" these clocks to keep coordinate time.

So the speed of light as measured via coordinate time changes with altitude, because of the way we adjust our coordinate clocks.

The underlying reason we need to adjust proper clocks in order to define a coordinate time standard is space-time curvature.

Talking about space-time curvature is rather abstract, but we can talk about the effects of curvature via a simple analogy of curvature of a more familiar sort. Specifically, we'll talk about the effects of the cuvature of the Earth on purely spatial measurements.

Imagine trying to locate a point on the surface of the Earth - which we will idealize as a sphere. You can specify a such a point by giving a longitude and a lattitude.

However, if you want to find the distance between two points on the surface of the Earth, you can no longer simply subtract the coordinates. A 1 degree difference in longitude will represent a different distance near the north pole than it will near the equator.

Suppose we ask "What is the effect of the curvature of the Earth on the speed of a sailing ship", as an analogy to the original question, "What is the effect of the curvature of space-time on the speed of light". The obvious answer to the first question is "none". But in order to come up with this answer of "none", you need to disregard the way that coordinates (such as longitude and lattitude) mislead you in computing distances, or perhaps become more sophisticated about how you compute distances (by using a metric, the same approach that GR uses - for instance see http://burro.cwru.edu/Academics/Astr328/Notes/Metrics/metrics.html) - and give primacy to non-coordinate based measurements of distance made by actual measuring instruments such as rulers - and in the space-time case, rulers and clocks.

This is actually the logical way to do things, and when you do things this way, the answer to the second question also becomes "there is no effect of space-time curvature on the speed of light".

However, the desire to use coordinate is so strong that people tend to, conceptually, dismiss the notion of time as measured by actual, physical, "proper clocks", and insist that the correct notion of the measurement of time is the one that fits into the coordinate time paradigm.

With the coordinate time paradigm (such as the specific example of TAI time), the speed of light does vary - but how it varies depends on the details of exactly what coordinate system you set up. Usually there are common choices in setting up a coordinate system, but they are by no means unique. For instance, the difference between an inertial observer and an accelerating observer is just such a choice of coordinates. So the speed of light becomes dependent on your coordinate choices, which makes it hard to talk about unambiguously, unless you exactingly define how you set up your coordinates, or, more commonly, briefly mention your choice of coordinates and hope that reader and author share the same detailed defintions.

 

[Naty1]

...the designers of the GPS system had to consider the possible effects of Shapiro delay (but determined experimentally that the delay was negligible).

I know there IS time correction required between Earth stations and satellites:

http://en.wikipedia.org/wiki/Gravitational_time_dilation#Experimental_confirmation

Gravitational time dilation has been experimentally measured using atomic clocks on airplanes. The clocks aboard the airplanes were slightly faster with respect to clocks on the ground. The effect is significant enough that the Global Positioning System's artificial satellites need to have their clocks corrected.[4]

but I don't know if that is considered 'Shapiro delay' or not:

http://en.wikipedia.org/wiki/Shapiro_delay#Calculating_time_delay

In a near-static gravitational field of moderate strength (say, of stars and planets, but not one of a black hole or close binary system of neutron stars) the effect may be considered as a special case of gravitational time dilation.The measured elapsed time of a light signal in a gravitational field is longer than it would be without the field, and for moderate strength near-static fields the difference is directly proportional to the classical gravitational potential, precisely as given by standard gravitational time dilation formulas.

In conjunction with pervect's description, you might also find this of interest [or not]
but I happened to recall GPS runs on its 'own time'...and that was updated last year [2012]


http://en.wikipedia.org/wiki/GPS_time#Timekeeping

 

[JVNY]

Thanks to all. I will respond to a number of the posts and then present an example (based on acceleration in SR) and hope that WannabeNewton or others will take it up and discuss it in GR terms (I am only beginning to learn the basics of GR, but hopefully the equivalence principle allows the SR example to illuminate something about GR).

The Shapiro delay is not currently incorporated into the GPS system, so it should not be captured by the references that Naty1 links. The signal propagation delay between Earth and satellites due to gravity is about 6 ps, which Ashby identifies as Shapiro delay. There is an additional 35 ps of cross link signal propagation delay due to gravity. Neither is incorporated yet. Here is a link on the effects, see slides 2 and 15: http://www.gps.gov/cgsic/meetings/2012/weiss2.pdf. Here is a piece explaining that the Shapiro delay was not incorporated into GPS because it was negligible: http://www.phy.syr.edu/courses/PHY312.03Spring/GPS/GPS.html .

Pervect's description of coordinate time accords with the analysis of coordinating clocks in a row of spaceships that accelerates Born rigidly (discussion of this at: http://en.wikipedia.org/wiki/Rindler_coordinates). But I do not see how this is relevant to the Shapiro delay.

The Shapiro experiment needs only one clock, a clock on the wrist of the experimenter. That clock records the elapsed time for a signal's out and back trip. That elapsed time is proper time; it is the wristwatch time of the watch on the experimenter's wrist.

I am not sure whether this involves "distant observation" as Naty1 mentions. There are only two observations: experimenter's wristwatch time when the signal departs, and experimenter's wristwatch time when the signal arrives back. Both observations are at the same place; neither is distant from the other. The light signal does travel a distance before reflecting and returning. Does this mean that the experiment measures the speed of light in a distant environment? Perhaps -- but every measure of the speed of light is an out and back measurement. There is no experiment that measures a one way speed of light.

Finally, the example. Say the Shapiro experimenter simultaneously sends signals in opposite directions, one toward the gravitational mass and the other away from that mass. In each case the flash travels the same distance then reflects and returns to the experimenter. The one that initially travels away from the mass will arrive first. It is not even necessary to utilize a clock or measure the speed of the signal. The fact that the forward signal arrives before the rearward one shows that signals traveling closer to the gravitational mass travel slower. If the experimenter measures the elapsed proper time for the two signals, the one that begins by traveling toward the mass will take a longer proper experimenter time to return than the one that begins by traveling away from the mass.

Attached is a diagram of three point mass spaceships beginning at rest in an inertial frame, then accelerating Born rigidly. The rear starts at x=0.5 and accelerates at proper rate 2; the center starts at x=0.75 and accelerates at proper rate a=1.33; the front starts at x=1 and accelerates at proper rate a=1. Upon beginning acceleration, the center ship simultaneously flashes light forward and back. The flashes reflect off of the other two ships and return to the center ship. The diagram shows four dashed lines of simultaneity for the row of ships at relevant points of the flashes' trips.

The forward flash returns first. Each leg of its trip (forward to the front ship, then back to the center) takes the same amount of proper center time (denoted Tc in the diagram) as the other leg (each leg takes Tc=0.216).

The rearward flash returns second. Each leg of its trip also takes the same amount of proper center time as the other leg (each takes Tc=0.304). The total proper center time for the round trips are Tc=0.432 for the forward flash and Tc=0.608 for the rearward flash.

The rearward light flash is slower than the forward light flash. This is not because of coordinate time. There is only one clock (the center ship clock), and it measures only its own proper time. Also, there is no need even to use a clock. All that the experiment requires is simultaneity; as long as the two flashes do not arrive simultaneously, then one is traveling slower than the other. You can change the rate of that clock in any way that you want to coordinate with any other clock in any other frame, and that cannot change the fact that the forward flash returns first; it is faster than the rearward flash.

There remains one important point, which is whether the distances between the ships stay constant during the acceleration; if they do not, then the conclusions above are not necessarily correct. By definition the distances remain constant in Born rigid acceleration. But the parts of Taylor and Wheeler in "Exploring Black Holes" that I have read so far explain that gravity is different in this regard -- the reduced radius around the mass is not the same as the radius. So perhaps it is the distance that is the issue, not the experimenter's proper clock rate.

 

[Dale]

In each case the flash travels the same distance

Are you sure about that? How are you defining the distance?

According to local rods and local clocks the light always moves at c, so the distance according to local rods along the path is longer for the path that takes longer. It sounds to me like you must be talking about some coordinate distance which is the same, in which case that can be made to be a true statement regardless of the details simply by picking the appropriate coordinates.

The fact that the forward signal arrives before the rearward one shows that signals traveling closer to the gravitational mass travel slower.

Yes, if the coordinate distances are equal and you are talking about coordinate speed. That is certainly true in your chosen coordinate system, but I can always transform to a coordinate system where it is false.

 

[Naty1]

from JVNY

Here is a link on the effects, see slides 2 and 15: http://www.gps.gov/cgsic/meetings/2012/weiss2.pdf.

Those are really interesting slides, the most detailed I have seen...

I don't really understand how all the separate effects are defined, but even
'tidal effects' has components:

Three Separate Contributions to Tidal
Effect on Clocks
1. Fractional frequency shift due to the tidal
potential itself
2. The tidal perturbation causes a change in
radius that contributes to the fractional
frequency shift
3. There is also a change in satellite speed

///////////////////
Slide #7:

Relativity and Clock Rates

Clocks run slower deeper in gravi-potential
Clocks run slower as they go faster

Seems like both these effects are accounted for in a single formula...on slide # 8, hadn't seen that before...

 

[JVNY]

I define distance in this example as the proper distance between the ships, that is the distance that the ships measure between themselves in their own reference frame. This is always 0.25. The ships start at rest in the inertial frame illustrated above at x=0.5, x=0.75, and x=1, hence the distance between each of them in their own frame is 0.25. Each then accelerates simultaneously, Born rigidly (at individual proper rates of acceleration that maintain their proper distance as measured in any momentarily comoving inertial frame). As wikipedia describes, "The defining property of Born rigidity is locally constant distance in the co-moving frame for all points of the body in question." http://en.wikipedia.org/wiki/Born_rigidity

Similarly, "Consider a uniform distribution of particles at rest along some segment of the x-axis of an inertial coordinate system x,t at the time t = 0. Each particle is subjected to a constant proper acceleration (hyperbolic motion) such that, with respect to its instantaneously co-moving inertial rest frames, the distances to each of the other particles remain constant. This condition is called 'Born ridigity' . . ." http://www.mathpages.com/home/kmath422/kmath422.htm

So, what the Minkowski diagram above shows is that in the proper reference frame of the three ships, and using the proper clock of the center ship, light travels more slowly when it travels in the area to the left of the center ship (and faster when it travels in the area to the right of the center ship). Both inputs to the calculation of speed are proper measures: proper distance between the ships in their own reference frame, and proper clock time on the the center ship's clock. But this example is in SR, not GR, and pervect refers above separately to distance (e.g., "local rulers"), and Naty1 refers above to space contracting.

 

[Dale]

Sorry, I guess I should have put more context into the quote. I was responding to your Shapiro delay comment. Not the ships.

EDIT: However, I guess that the question about how you are defining distance does apply to both the Shapiro delay scenario and the accelerating ships scenario. What you have shown with the ships is the well-known fact that the coordinate distance is not the same as the "radar distance" in the Rindler chart. So, again, the coordinate speed of light depends on the definition of your coordinates. The same thing happens in other non-inertial coordinate systems, such as rotating reference frames where the coordinate speed of light can be made arbitrarily fast or slow.

I am not sure if you are asking any specific question with your examples, or simply working through the concepts on your own.

 

[JVNY]

Thanks. I am trying to understand the terminology. In SR the texts refer to proper distance (in the object's own frame) and length contracted distance (as the object presents in other frames in inertial movement with respect to the object). Posters in this thread refer to local, coordinate and radar measurements, and it is not clear to me what exactly they refer to. For example, what is coordinate distance v. radar distance in the Rindler chart, and how do they compare to proper distance?

 

[WannabeNewton]

Radar distance is a very general method of establishing a notion of distance between observers in a congruence of time-like worldlines within an arbitrary space-time.

The idea is as follows: observer $O$ in the congruence sends a light signal at an event $p_1$ to observer $O'$ in the congruence who immediately reflects the signal back to $O$ at event $p_2$ whereupon $O$ receives the signal at event $p_3$; $O$ records the time $\Delta \tau$ between events $p_1$ and $p_3$ on a clock that he/she carries (i.e. the proper time along the worldline of $O$ between $p_1$ and $p_3$).

The radar distance between $O$ and $O'$ is then $L = \frac{c\Delta \tau}{2} = \frac{\Delta \tau}{2}$ where I have set $c = 1$.

One can easily compute this in Rindler space-time: the metric is $ds^2 = - x^2 dt^2 + dx^2$. The congruence of Rindler observers is given by the vector field $\xi^{\mu} = \frac{1}{x}(\partial_t)^{\mu}$ i.e. the Rindler observers are exactly the static observers in Rindler space-time.

Now consider two Rindler observers $O$ and $O'$, with $O$ at some $x_0$ and $O'$ at some $x$. A null geodesic (light signal) originating at $x_0$ at time $t_0$ on the worldline of $O$ is given by (after a simple integration) $ds^2 = 0 \rightarrow t - t_0 = \ln x - \ln x_0 = \ln \frac{x}{x_0}$. The light signal reflected by $O'$ clearly has the exact same form so the total coordinate time between emission and reception by $O$ is then $\Delta t = 2\ln \frac{x}{x_0}$ hence the proper time between emission and reception as read by a clock carried by $O$ is given by $\Delta \tau = x_0 \Delta t = 2x_0 \ln \frac{x}{x_0}$ therefore the radar distance between $O$ and $O'$ is $L = x_0 \ln \frac{x}{x_0}$.

On the other hand the local ruler distance is given by $dl = [(g_{\mu\nu} + \xi_{\mu}\xi_{\nu})dx^{\mu}dx^{\nu}]^{1/2}$.

Local ruler distance has the following interpretation: say we have an event $p_1$ on the worldline of our Rindler observer $O$ and another event $p_2$ that is infinitesimally separated from $p_1$. $O$ can connect these two events using a displacement vector (also called a separation vector) $dx^{\mu}$; if $O$ wants to find the spatial distance between these two events then $O$ has to project $dx^{\mu}$ onto his/her local simultaneity slice and then take the length of this projection.

You can think of the projection of $dx^{\mu}$ onto the local simultaneity slice of $O$ as an infinitesimal ruler extending from $O$ at event $p_1$ to the event $p_2$. Mathematically the projection is given by $\eta^{\mu} = dx^{\mu} + (\xi^{\gamma}dx_{\gamma})\xi^{\mu}$
Note that this is just the orthogonal projection since the local simultaneity slice can be shown to coincide with the orthogonal projection at $p_1$.

A simple calculation then shows that $(\eta^{\mu}\eta_{\mu})^{1/2} = dl$ so we are indeed finding the length of the ruler represented by $\eta^{\mu}$.

Now $\xi_{\mu} = -x(dt)_{\mu}$ so $dl = [g_{\mu\nu}dx^{\mu}dx^{\nu} + x^{2}dt^{2}]^{1/2} = dx$.

If we compute the radar distance between these two infinitesimally separated events then we get $dL = x_0 \ln (1 + \frac{dx}{x_0}) = dx$, where I have Taylor expanded the ruler distance to 1st order.

So for nearby events the ruler distance and radar distance agree. However it is clear from the above discussion that for events that are not nearby (i.e. not infinitesimally separated), radar distance will not agree with ruler distance.

EDIT: In finding the radar distance, I assumed that the coordinate time for the light signal to go from $p_1$ to $p_2$ equals its coordinate time to go from $p_2$ to $p_3$; this always works if the space-time is static, which Rindler space-time is.

 

[Dale]

For example, what is coordinate distance v. radar distance in the Rindler chart, and how do they compare to proper distance?

In flat spacetime you can define the proper distance between two events or the proper distance between two worldlines at rest wrt each other. The proper distance between two spacelike events is the distance in the inertial frame where they are simultaneous. The proper distance between two worldlines is the distance between them in the inertial frame where the worldlines are both at rest.

In curved spacetime these notions no longer exist, however you can define the proper distance along a spacelike path as $\int \sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}$.

 

[pervect]

OK, thanks. It would still be great if pervect would explain more about the coordinate v. local clocks (if still available and willing) because coordinate time does seem to be important. For example, Neil Ashby has stated that GPS time is an example of coordinate time, and I have read that the designers of the GPS system had to consider the possible effects of Shapiro delay (but determined experimentally that the delay was negligible).

I'm not sure what more I can explain. Coordinate clocks are defined by the coordinate system - if you change the coordinate system, you change the coordinate clocks. Thus they are a matter of convention. In general they don't have much physical significance.

Proper time represents the readout of a ideal clock following a specific worldline. A stationary, orbiting, or otherwise moving atomic clock would be an experimental measurement of proper time along the clocks worldline, at least up to the experimental accuracy of atomic clocks.

I hope the following observations will be helpful:

Being able to navigate, measure distances, and plan long trips on a sphere, such as the Earth, requires some knowledge of spherical geometry and to do the calculations, spherical trignometry. Explanations based in planar geometry just aren't going to be adequate.

Being able to navigate, measure distances, and plan long trips (including plotting the paths of radar signals over long distances) in a curved space-time, requires some knowledge of the Riemannian geometry of General Relativity.

If you want to travel short distances on the Earth, for instance if you want to walk to your local store, and you don't mind minuscule discrepancies in the measurements, you CAN "get away" without using spherical geometry - you can regard the Earth's surface as "locally flat", and draw a flat map on the plane, and get around OK. You don't need spherical geometry to plan a trip to your local store. However unless your accuracy requirements are very lax indeed, you do need spherical geometry to plan a trip from New York to London.

There is a way to do this in GR, as well.

In a small area, where you can approximate curved space-time as a flat plane, the coordinates with arguably the most significance are Fermi Normal coordinates. They are like a "flat map" of the surface of the Earth, as far as navigating through space-time.

A somewhat helpful reference (which does better at talking about the qualities of Fermi Normal coordinates rather than defining them in understandable terms):

http://arxiv.org/abs/arXiv:gr-qc/9402010

A very convenient property of flat space-time is the existence of Cartesian coordinates which are directly related to the experience of an inertial observer, and which are easily interpreted in terms of physical quantities like, e.g., distances. This concept should, in some sense, extend to curved space-time since actual observers are often accelerated and live in a (weakly) curved space.

There is a general agreement that Fermi normal coordinates are the natural coordinate system of an observer in general space-times. By this we mean coordinates which are most closely to Cartesian coordinates for general relativistic situations and which are related to observable quantities like acceleration, rotation, or distances. They were first introduced by M. Fermi in 1922

The paper's presentation of how you generate Fermi normal coordinates is probably too technical for the average reader, however. Even the simplest presentation is going to require some familiarity with the concepts of geodesics. (I'm not aware of any reasonably simple published exposition of them, and attempts in the past to present them with forum posts seem to have floundered because required concepts like geodesics are unfamiliar into too many readers).

A few other comments in a different vein:

A few lines from the reference
http://www.gps.gov/cgsic/meetings/2012/weiss2.pdf
quoted earlier caught my eye:

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Coordinate speed of light is generally slower than c due to Earth’s gravitational field
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The length of a light path is the coordinate length plus the so-called Shapiro delay

This is pretty much the same thing I've been saying. The coordinate speed of light does , in fact, vary. This is true, but the remark has little physical significance because coordinate speeds have little physical significance.

An interesting point is the statement that they get a length by adding the shapiro time delay multiplied by "c" to the coordinate length. They don't justify this in detail, but it seems to be quite consistent with the viewpoint that the shapiro delay can be logically viewed in terms of the distance changing, rather than the speed of light changing.

 

[JVNY]

This is great, thanks to all. A lot to read and work through. It does seem that distance is changing, and I will work through the various measures of distance.