Speed of light exceeded??

[peteshimmon]

Now now Strange, you are using that simple
equator analogy to try and flummox me. I know
an equatorial line is curved because I can see
it going below the horizon.

Light follows a geodesic path in a gravitational
field. It has been proved beyond all doubt. a
light photon follows the geodesic. Whether
particles with very small mass moving as near
light speed as makes no difference follows
the same path as light has not been demonstated.

Now suppose for the hell of it someone calculated
the notional difference in path lengths for a
straight line and the geodesic past a gravitating
body. And this extra length would account for
half the light delay. Would that be illustrating
something?

[HenrikOlsen]

Which difference between the straight line and the geodesic? The geodesic IS the straight line.

If you're thinking the line that looks like it's straight when seen from outside, it's longer than the geodesic.

[Jeff Root]

Shapiro delay (as mentioned by Pete): The additional time
required for light to go from point A to point B because a large
mass between points A and B forces the light to take a curved
path rather than a straight line. The curved path is longer, so
the light is delayed when the mass is present.

-- Jeff, in Minneapolis

[Strange]

Now now Strange, you are using that simple
equator analogy to try and flummox me. I know
an equatorial line is curved because I can see
it going below the horizon.

But the equator is still the shortest distance between two (equatorial) points. The shortest distance between two points on the surface of the earth is a great circle or geodesic[1], the equivalent of a straight line on a curved surface.

Light follows a geodesic path in a gravitational
field. It has been proved beyond all doubt. a
light photon follows the geodesic. Whether
particles with very small mass moving as near
light speed as makes no difference follows
the same path as light has not been demonstated.

Anything moving in a "straight line" (in the Newtonian sense - with no force[2] acting on it) will travel along a geodesic. So neutrinos will follow exactly the same path as photons. In curved space, taking anything other than the geodesic path will take longer.

Now suppose for the hell of it someone calculated
the notional difference in path lengths for a
straight line and the geodesic past a gravitating
body. And this extra length would account for
half the light delay. Would that be illustrating
something?

No because the "straight line" is the geodesic.

[1] In fact "geodesic" comes from geodesy, the science of measuring the earth.
[2] Real forces, that is; i.e. not gravity.

[caveman1917]

Suppose you are at some position on a curved manifold, and you decide to start walking. At each point you pass, you just go straight on (locally any curved manifold will be flat, so at each point you can make the intuitive local decision of going straight). The resulting path is a geodesic, which is a straight line (you went straight ahead the entire time). It doesn't look straight when you project it onto a flat coordinate system, but that's just because you're projecting it.

As Strange said, if you for example do this on the surface of the earth (start walking and keep walking straight all the time), you'll get a great circle. It will look curved on a map (check a map of airline paths for example), but that's only because you're projecting onto a flat map, and IIRC there are projections that don't do that.

In any case, a geodesic is defined as the resulting path when you locally make the decision to keep going straight on at each point you pass, how is that not a straight line?

ETA: that's also why it's the path followed by a particle without any forces acting on it, if there were a force acting on it at any point along the path, it wouldn't go straight on anymore (since a force is a change in motion).

[Jeff Root]

Suppose you are at some position on a curved manifold, and
you decide to start walking. At each point you pass, you just go
straight on (locally any curved manifold will be flat, so at each
point you can make the intuitive local decision of going straight).
The resulting path is a geodesic, which is a straight line (you
went straight ahead the entire time). It doesn't look straight
when you project it onto a flat coordinate system, but that's just
because you're projecting it.

As Strange said, if you for example do this on the surface
of the earth (start walking and keep walking straight all the time),
you'll get a great circle. It will look curved on a map (check a
map of airline paths for example), but that's only because you're
projecting onto a flat map, and IIRC there are projections that
don't do that.

Projection is irrelevant. If the path is curved it is curved.
It will look curved from at least some viewing angles, and in
at least some projections. It may or may not look straight
from some viewing angles and in some projections.

(Of course, any illustrations of such a path on a computer
screen will necessarily be 2-D projections, but we are able
to ignore that and visualize a path in 3-D space. 4-D, not
so much.)

In any case, a geodesic is defined as the resulting path when
you locally make the decision to keep going straight on at each
point you pass, how is that not a straight line?

It is not a straight line in exactly the way you just very clearly
described. (Aside from use of the jargon term "manifold" which
probably has a meaning understood by some people.)

-- Jeff, in Minneapolis

[Jeff Root]

What would the Shapiro delay be for light travelling straight
through a gravitationally-lensing object rather than around it?

-- Jeff, in Minneapolis

[Strange]

Projection is irrelevant. If the path is curved it is curved.
It will look curved from at least some viewing angles, and in
at least some projections. It may or may not look straight
from some viewing angles and in some projections.

Your last sentence seems to contradict the first; how can projection be irrelevant if in some projections it looks straight and some it looks curved.

Also, on the 2D surface itself, the line will not look curved (it will follow the surface in a straight line). Your talk of "viewing angles" is a projection.

[HenrikOlsen]

It is not a straight line in exactly the way you just very clearly
described. (Aside from use of the jargon term "manifold" which
probably has a meaning understood by some people.)

It IS a straight line in exactly that sense. Which means that for something moving IN that curved space, there are no turns at any time, there's just moving straight ahead by the only criterion possible for something in that space, which within that space is straight ahead at all time. That it looks curved from outside that space is a result of looking at it through a projection between spaces with different curvatures.

This isn't physics, it's mathematics, it's a fundamental property of curved spaces.

Trying to find a path that is shorter than the geodesic requires a path that leaves the space completely and enters again somewhere else. And claiming that neutrinos would leave space-time and could manage to return again at the point of the detector requites quite a lot of phlebotinum.

[Jeff Root]

It is not a straight line in exactly the way you just very clearly
described. (Aside from use of the jargon term "manifold" which
probably has a meaning understood by some people.)

It IS a straight line in exactly that sense. Which means that
for something moving IN that curved space, there are no turns
at any time, there's just moving straight ahead by the only
criterion possible for something in that space, which within
that space is straight ahead at all time. That it looks curved
from outside that space is a result of looking at it through a
projection between spaces with different curvatures.

As I said before, if it is not curved, but straight, then call
it "straight" and quit using the useless term "geodesic".

This isn't physics, it's mathematics, it's a fundamental
property of curved spaces.

Certainly.

Although looking at actual physical things and their properties,
such as Shapiro delay, can help give meaning to the abstract
and meaningless mathematics.

Trying to find a path that is shorter than the geodesic requires
a path that leaves the space completely and enters again
somewhere else. And claiming that neutrinos would leave
space-time and could manage to return again at the point
of the detector requites quite a lot of phlebotinum.

The neutrinos supposedly take a very slightly curved path
through the Earth because of Earth's gravity. That path is
longer than a straight line would be in "the same place" if
the Earth weren't there. Light would take that same path
if the neutrinos are massless, or approximately the same
path if they have mass.

Gravitational lensing and Shapiro delay of light have been
measured. Gravitational lensing and Shapiro delay of
neutrinos have not been measured.

-- Jeff, in Minneapolis

[Jeff Root]

Projection is irrelevant. If the path is curved it is curved.
It will look curved from at least some viewing angles, and in
at least some projections. It may or may not look straight
from some viewing angles and in some projections.

Your last sentence seems to contradict the first; how can
projection be irrelevant if in some projections it looks
straight and some it looks curved.

There is no contradiction. Caveman said:

a great circle ... will look curved on a map ..., but that's
only because you're projecting onto a flat map,

Projection of a great circle onto a map is not the only
reason it looks curved. It also looks curved because it
*is* curved. The specific projection chosen determines
whether the great circle's curvature is revealed or hidden,
and whether spurious curvature is introduced or not.

The sides of a building will look curved in a photo, but that's
only because you're projecting the building onto a flat sheet.

http://www.cityprofile.com/forum/att...art-museum.jpg

The sides are actually parallel. It's just the projection that
makes them look curved. Trust me.

Also, on the 2D surface itself, the line will not look curved
(it will follow the surface in a straight line).

Whether it looks curved or not depends on the viewing
angle.

Your talk of "viewing angles" is a projection.

Yes, viewing angle is a subset of projections.

-- Jeff, in Minneapolis

[Strange]

There is no contradiction. Caveman said:

And he went on to say that there were other projections that would keep great circles straight (although a 2D map can only have some geodesics appear straight, as far as I know). And then you went on to describe other projections that give the same effect (or not). So the projection is relevant.

Projection of a great circle onto a map is not the only
reason it looks curved. It also looks curved because it
*is* curved.

Well, I would say it only looks curved when viewed in (projected to) a higher dimensional space. But that isn't really relevant. You can consider it curved. But it is still the equivalent of a straight line in that it defines the shortest distance between two points. You can't say that if things didn't follow the geodesic they would take a shorter path because the geodesic is defined to be the shortest path.

[Strange]

As I said before, if it is not curved, but straight, then call
it "straight" and quit using the useless term "geodesic".

How would you define a "straight line" between two points on the surface of the earth? As you have already said, such a line will be curved. Maybe we need a name for such a path, rather than having to constantly refer to "the shortest distance between two points on the curved surface". Hmmm... why don't we call it a "geodesic" as we often use them when measuring the earth...

[Jeff Root]

If the line on the surface of the Earth is straight, then call it
a "straight line". If it is not straight, but instead is curved,
then call it a "geodesic".

-- Jeff, in Minneapolis

[Strange]

But what do you mean by straight? Is a line of longitude straight? Is a line of latitude straight?

[Jeff Root]

There is no contradiction. Caveman said:

a great circle ... will look curved on a map ..., but that's
only because you're projecting onto a flat map,

And he went on to say that there were other projections
that would keep great circles straight (although a 2D map
can only have some geodesics appear straight, as far as I
know). And then you went on to describe other projections
that give the same effect (or not). So the projection is
relevant.

The projection is relevant to whether the lines on the map
are curved or not. The projection is irrelevant to whether
great circles are curved or not. The question is whether
great circles (proxies for light paths) are curved or not.
I don't care whether their projections on maps are curved.
That is irrelevant to the question.

Projection of a great circle onto a map is not the only
reason it looks curved. It also looks curved because it
*is* curved.

Well, I would say it only looks curved when viewed in
(projected to) a higher dimensional space.

In other words, it looks curved when you look at it
normally instead of just looking at it from a sweet
spot which gives the illusion of it being straight.

-- Jeff, in Minneapolis

[Strange]

On the 2D surface it doesn't look curved.

In our 3+1D world a geodesic doesn't look curved. So your definition of normal seems to mean stepping outside the universe.

[Jeff Root]

On the 2D surface it doesn't look curved.

Sure it does!

Lay a perfectly flat board on a perfectly "level" surface and
you'll see that the board wobbles like a teeter-totter on the
curved "perfectly level" surface.

Watch the masts on a ship as it sails away from you and
you'll see them tip over farther and farther as it disappears
over the horizon.

Go to the equator and take a look at the line on the ground,
and you'll see that the highest point is right in front of you,
and it curves downward to the east and west.

The horizon isn't a straight line.... It is a circle that goes
all the way around me. I'm in the center!

In our 3+1D world a geodesic doesn't look curved.

Sure it does!

When I throw snowballs, they don't fly up into the sky and
disappear.... They fly up a few feet, arc over, and fall back
down to the ground.

The International Space Station goes around in circles.

The paths of light going past and through Abell 2218 are
curved toward me -- ME! If they didn't curve, I wouldn't
be able to see the light!

http://apod.nasa.gov/apod/image/0110/a2218c_hst_big.jpg

So your definition of normal seems to mean stepping
outside the universe.

You're the one suggesting that idea. It doesn't sound
normal to me.

-- Jeff, in Minneapolis

[Strange]

Watch the masts on a ship as it sails away from you and
you'll see them tip over farther and farther as it disappears
over the horizon.

Maybe we are taking at cross-purposes. We are not talking about the curvature of the surface (or at least I wasn't) but about the nature of "straight" lines on that surface. If the ship is taking the shortest route on this curved surface, then it will be following a geodesic - a curved straight line.

Is your objection just to the use of a specialised term for the (generic) shortest distance between two points? Or do you think there is a shorter distance than the geodesic/straight-line route?

I'm no longer sure what we are arguing about and we have got totally off-topic, so perhaps we should drop it there.

[Ara Pacis]

I think the confusion is that the term geodesic is used both for mathematics and for the physical properties of earth. We know that the concept of the surface of the earth being curved is geodesy, but that meaning of geodesic seems divorced from the mathematical space-time meaning since the flightpath of the neutrinos didn't actually follow the curvature of the earth but took their own path through it. In other words, we *can* tunnel through the earth, so using the earth as an example of space-time geodesy is a bad example, even if it was the inspiration of the term.

[peteshimmon]

I seem to have caused this debate with my
speculation about the paths neutrinos might
follow. Some of you have no doubt they go
the same way light goes. Fair enougth. But
the 60 nsec discrepancy looks a fact and
thoughts about things connected with the
Shapiro delay seems a natural train of thought.

The paths the neutrinos follow through the
rock should still be influenced by gravity
and if light could go the same route it would
have about a 10 necs delay. I expect this was
included in the calculations at CERN.

[HenrikOlsen]

In other words, it looks curved when you look at it
normally instead of just looking at it from a sweet
spot which gives the illusion of it being straight.

I think I found where we talk past each other, it's in our choices of which way of looking at it is "normally".

To me, the relevant spots from which to look at the line are the points on the line itself, every other spot has the potential of giving a distorted impression of what's going on with the line.

"Seen from outside" is an arbitrary sweet spot which can distort what is going on.

A great circle is straight in the sense that matters for people travelling along it.

[HenrikOlsen]

I seem to have caused this debate with my
speculation about the paths neutrinos might
follow. Some of you have no doubt they go
the same way light goes. Fair enougth. But
the 60 nsec discrepancy looks a fact and
thoughts about things connected with the
Shapiro delay seems a natural train of thought.

The paths the neutrinos follow through the
rock should still be influenced by gravity
and if light could go the same route it would
have about a 10 necs delay. I expect this was
included in the calculations at CERN.

No, we expect them to have mass and therefore take a path slightly different from the one take by light, this path will have been slightly longer because light follows the geodesic (aka straight line) which is the shortest path there is within that space. For the neutrinos to follow a shorter path they would have to leave the universe and know to come back right at the detector.

[caveman1917]

Projection is irrelevant.

Strictly speaking yes it is indeed irrelevant. I only brought it up because i thought that might be the underlying reason why you consider a geodesic curved, the intuition from projecting things onto a flat space (which we always intuitively do).

If the path is curved it is curved.

What would be your definition of "curved"?

I only know of one definition that could be relevant (intrinsic curvature isn't defined for a 1-dimensional space such as a line), and that is to consider a line curved whenever it is not a geodesic. So it is clear you're using another definition and without knowing what each person understands under the terms they use, this discussion could go on ad infinitum

(Aside from use of the jargon term "manifold" which
probably has a meaning understood by some people.)

It's a topological space that locally approximates euclidean (or in our case minkowski) space at each point. This property is what allows you to extend the concept of "straight", since you can define it as the result of locally going straight in the normal (euclidean/minkowskian) sense at each point, which is exactly how a geodesic is defined.

[caveman1917]

And he went on to say that there were other projections that would keep great circles straight (although a 2D map can only have some geodesics appear straight, as far as I know).

That's probably true, i wasn't entirely sure about that statement i made.

[caveman1917]

No, we expect them to have mass and therefore take a path slightly different from the one take by light, this path will have been slightly longer because light follows the geodesic (aka straight line) which is the shortest path there is within that space.

They both will follow geodesics. Remember that we are using spacetime, not just space, as the background in which to define geodesics. So for example if i shoot up a projectile and it comes back to me, the projectile will have followed the shortest path between the two events of "me shooting" and "me getting hit on the head by projectile", i'll have taken a longer path between those events even though if you just look at it spatially i remained in the same place all the time.

The difference with light and massive objects is that light will follow null-geodesics and massive objects will follow timelike geodesics. The arclength of a path followed by light will be zero (hence null geodesic), where for a massive object it will be positive. So the path taken by light really is the shortest path possible, it has length zero, to go faster than that your path must have negative length.

[HenrikOlsen]

Oops, yes, sorry.

I was thinking light only, which is where the geodesic and the straight line are identical. For particles with mass, the geodesic is indeed not the "locally straight line" of light, but is always longer.

If you're looking at "shortest path" as the path resulting in shortest time experienced by the travelling thing, then yes, neutrinos do follow the shortest path for them, which is slightly different, and longer, than the straight line path taken by light.

[caveman1917]

Oops, yes, sorry.

I was thinking light only, which is where the geodesic and the straight line are identical. For particles with mass, the geodesic is indeed not the "locally straight line" of light, but is always longer.

If you're looking at "shortest path" as the path resulting in shortest time experienced by the travelling thing, then yes, neutrinos do follow the shortest path for them, which is slightly different, and longer, than the straight line path taken by light.

I meant it differently. Massive object will follow geodesics (straight lines), the null-geodesics followed by light aren't so much geodesics since they have length zero. It's like asking (purely spatially), if i go from point p to point p, did i go in a straight line? The answer could be "yes" since you didn't take a turn anywhere, but it's not like you really took any path, the path has no length. In other words, if a point is a line would it be straight? It's only in that sense that null-geodesics are geodesics. When saying the geodesic between two events (in the sense we've been talking about it, ie as a "straight line" that actually is a line) it is implied that it is for massive objects.

But on second thought this entire subdiscussion about geodesics is next to the point. The problem with something arriving before light (in the sense of the experiment) has nothing to do with its path being a geodesic or not. For one because we can only compare path lengths between the same set of events and if something arrives before something else, the "arrival events" will be different. Though i suppose we could say that it is a matter of arclength of the path, since arclength gives proper time, and for something to arrive before light its path must give it a negative proper time interval (ie the neutrino's clock must have been running backwards) irrespective of wether the path is a geodesic or not.

If you're looking at "shortest path" as the path resulting in shortest time experienced by the travelling thing

Counterintuitively it is the other way around (there is a minus sign there), the geodesic (shortest path) maximizes proper time.

[pzkpfw]

A brief mod note: Jeff Root et al, if you wish to argue against the commonly understood (in the context of this thread) meanings of things such as "curved space", "straight lines" and "geodesics", please do it in another place, such as in an ATM thread. caveman1917, Strange et al are using mainstream definitions here.

http://en.wikipedia.org/wiki/Geodesic , http://www.jb.man.ac.uk/~jpl/cosmo/metric.html , etc

[Jeff Root]

Maybe we are taking at cross-purposes. We are not talking
about the curvature of the surface (or at least I wasn't) but
about the nature of "straight" lines on that surface.

The two seem to me to be inseperably connected.

If the ship is taking the shortest route on this curved surface,
then it will be following a geodesic - a curved straight line.

Is your objection just to the use of a specialised term for the
(generic) shortest distance between two points?

The way you worded this question makes it ambiguous.
When you say "specialized term" it looks like you must be
referring to the term "geodesic". I have no objection at all
to use of the term "geodesic", despite my repeatedly telling
you not to use it. When I've said that, I always preceeded
the directive with an "if". What follows the "if" is the bit that
should concern you:

If a great circle is not curved, but straight, then call it
"straight" and quit using the term "geodesic".

A curved line is not a straight line. All curved lines have
some of the same properties that straight lines have.
Geodesics, in particular, tend to have more properties in
common with straight lines than other curved lines have.
But geodesics are NOT necessarily straight lines! The
assertion that they ARE straight lines is what I object to.

Again: Geodesics have many properties in common with
straight lines. But a geodesic is not usually a straight line.
In particular, a great circle is not a straight line!

Or do you think there is a shorter distance than the
geodesic/straight-line route?

Obviously. It is the reason why we are talking about it.
The straight line between two points on Earth's surface is
shorter than the great circle geodesic between those two
points. The geodesic is curved because it is on Earth's
surface. The straight line goes through the Earth.

I'm no longer sure what we are arguing about

I have been arguing that you should not call a curved
geodesic "a straight line". Real straight lines feel insulted
by that confusion.

-- Jeff, in Minneapolis
.