# The Direction of geodesics

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### Luigi Fortunati

Sep 7, 2022, 2:22:46 AM9/7/22
to
A geodesic passes from A and also from B.

Is the direction from A to B fully equivalent to the direction from B
to A?

Or can it happen that one of the two directions prevails over the
other?

### Richard Livingston

Sep 7, 2022, 1:12:17 PM9/7/22
to
In general relativity, the physical (spatial) distance from A to B is always
the same as from B to A. This is not true of the time for light travel
however. If A is higher in a gravitational field than B, the round trip
light travel time is longer for A to B to A (from the point of view of A)
than it is for B to A to B (from the point of view of B). For any single
observer the round trip times are the same either way. That may seem
paradoxical, but it is actually true and consistent.

Rich L.

[[Mod. note -- Referring to Richard's opening sentence, I take it that
he is referring to the metric distance integrated along the geodesic
between points A and B. This is also known as the "geodesic distance".
-- jt]]

### Luigi Fortunati

Sep 7, 2022, 3:43:54 PM9/7/22
to
Richard Livingston mercoledě 07/09/2022 alle ore 05:12:12 ha scritto:
>> A geodesic passes from A and also from B.
>>
>> Is the direction from A to B fully equivalent to the direction from B
>> to A?
>>
>> Or can it happen that one of the two directions prevails over the
>> other?
>
> In general relativity, the physical (spatial) distance from A to B is always
> the same as from B to A. This is not true of the time for light travel
> however. If A is higher in a gravitational field than B, the round trip
> light travel time is longer for A to B to A (from the point of view of A)
> than it is for B to A to B (from the point of view of B). For any single
> observer the round trip times are the same either way. That may seem
> paradoxical, but it is actually true and consistent.

I absolutely agree with what you wrote.

But how does all this affect the direction of motion that a body
decides to take when it is left free to follow its geodesic?

I simplify with my animation
https://www.geogebra.org/m/zdevssyz
where there is an elevator that (initially) cannot follow its geodesic
since it is constrained at point A.

If we eliminate the constraint (Start) the elevator starts moving
towards B to follow its geodesic which I have highlighted with the blue
dotted line.

This blue line has no limits and has no privileged directions, it is
not a vector that goes in an obligatory direction.

One direction is as good as the other.

But the elevator always goes in one direction only, the one that goes
down, towards point B.

And it never goes up.

What is it that forces the elevator to always get started downwards and
never upwards?

What has the down direction more than the up direction?

In other words, are geodesics open lines or are they vectors with one
direction privileged over the other?

### Stefan Ram

Sep 7, 2022, 11:41:23 PM9/7/22
to
Luigi Fortunati <fortuna...@gmail.com> writes:
>Richard Livingston mercoledi 07/09/2022 alle ore 05:12:12 ha scritto:
>>In general relativity, the physical (spatial) distance from A to B is always
>>the same as from B to A. This is not true of the time for light travel
>>however. If A is higher in a gravitational field than B, the round trip
>>light travel time is longer for A to B to A (from the point of view of A)
>>than it is for B to A to B (from the point of view of B). For any single
>>observer the round trip times are the same either way. That may seem
>>paradoxical, but it is actually true and consistent.
>I absolutely agree with what you wrote.

Richard wrote about two points in /space/.

In GR, a geodesic usually is a curve in /space-time/.

It's points are not (x, y, z), but (t, x, y, z).
One should not think of its end points as being two points in space!

A geodesic has no direction by itself.

### Luigi Fortunati

Sep 8, 2022, 6:36:03 PM9/8/22
to
Stefan Ram mercoledě 07/09/2022 alle ore 15:41:17 ha scritto:
> Richard wrote about two points in /space/.
>
> In GR, a geodesic usually is a curve in /space-time/.
>
> It's points are not (x, y, z), but (t, x, y, z).
> One should not think of its end points as being two points in space!

Okay, let's talk about two events in spacetime.

Event A is the lift's departure from point A and event B is the lift's
arrival at point B.

Why does the elevator (when it starts) go down (i.e. it goes from event
A to event B) and doesn't go up (where event B is not there)?

> A geodesic has no direction by itself.

On this I absolutely agree, however the elevator of my animation
https://www.geogebra.org/m/zdevssyz
it goes only and always in one direction and never in the other, why?

### Richard Livingston

Sep 9, 2022, 3:19:19 AM9/9/22
to
On Wednesday, September 7, 2022 at 2:43:54 PM UTC-5, Luigi Fortunati wrote:
> Richard Livingston mercoledÄ=9B 07/09/2022 alle ore 05:12:12 ha scritto:
> >> A geodesic passes from A and also from B.
> >>
> >> Is the direction from A to B fully equivalent to the direction from B
> >> to A?
> >>
> >> Or can it happen that one of the two directions prevails over the
> >> other?
> >
> > In general relativity, the physical (spatial) distance from A to B is always
> > the same as from B to A. This is not true of the time for light travel
> > however. If A is higher in a gravitational field than B, the round trip
> > light travel time is longer for A to B to A (from the point of view of A)
> > than it is for B to A to B (from the point of view of B). For any single
> > observer the round trip times are the same either way. That may seem
> > paradoxical, but it is actually true and consistent.
> I absolutely agree with what you wrote.
>
> But how does all this affect the direction of motion that a body
> decides to take when it is left free to follow its geodesic?
>
...
> What is it that forces the elevator to always get started downwards and
> never upwards?
>
> What has the down direction more than the up direction?
>
> In other words, are geodesics open lines or are they vectors with one
> direction privileged over the other?

There are many geodesics passing through any given point (event) in
space-time. What you are asking is how does a velocity 4-vector evolve.
The velocity 4-vector for a "stationary" object is (c, 0, 0, 0) in flat
space-time. In the vicinity of a large mass M it is (c(1-2M/r)^-1/2, 0,
0, 0) (ref eqn. 20.58 in "Gravity: an introduction to Einsteins general
relativity" by J. B. Hartle, 2003). The most rigorous way to calculate
this is to calculate the covariant derivative in the metric you are
concerned with. For the Schwarzschild metric this gives an acceleration

(Revtex code:
a^\alpha = \GAMMA^{\alpha}_{tt} \left( u^t \right)
)

where \GAMMA is a Christoffel symbol, which gives
a = (0, M/r^2, 0, 0)

This is actually the acceleration necessary to hold the object
stationary at this radius. If the object was released it would be in
free fall (i.e. no acceleration in the objects frame). In terms of the
stationary observer the free falling object would accelerate at a = (0,
-M/r^2, 0, 0) i.e. a radial acceleration towards the mass M.

the covariant derivative, while precise and correct, is almost opaque to
intuitive understanding of the physics, which I believe is what you are

The covariant derivative is related to the concept of parallel
transport. If we introduce a simple idea from quantum mechanics we can
more easily visualize the parallel transport of the particle wave
function in time, and recognize the resulting acceleration towards the
mass.

In relativity a particle with mass m has energy mc^2. In QM this is a
frequency mc^2/h, which I will call f = 1/T. For an object at rest, by
definition the wave function is in phase over some local region of
space. In terms of flat space-time the phase of this wave function
advances by 2pi every time interval T. This happens over the local
region at the same time as the g_{00} metric component is a constant
everywhere in flat space-time.

However near a large mass M, due to the gravitational red shift, not
every location has the same frequency. A region closer to the mass M
will appear to advance in phase more slowly, and a region further out
will advance more quickly. As a result the wave function for a
stationary object of mass m, while initially in phase over some local
area, will gradually become more and more out of phase along a radial
axis. This is entirely because the g_{00} metric component varies along
the r coordinate direction. The change in phase along r is, in QM, the
momentum of the particle, and this momentum gradually increases with
time. It is easy to show that this gives the same answer as the
covariant derivative.

Does that help?

Rich L.

### Stefan Ram

Sep 9, 2022, 4:30:05 PM9/9/22
to
Luigi Fortunati <fortuna...@gmail.com> writes:
>it goes only and always in one direction and never in the other, why?

I currently can't go on websites, but maybe what is intended
is this: a particle is placed on a geodesic compatible with
that particle. The particle now can move on the geodesic,
but how does it know in which direction as the geodesic has
no preferred direction?

The answer might be: While a geodesic has no preferred
direction, the time coordinate has. The particle moves into
that direction where its time coordinate is growing - assuming
the time coordinate was chosen in such a natural manner that
it grows towards the future of the given universe.

### Luigi Fortunati

Sep 9, 2022, 4:30:57 PM9/9/22
to
No, unfortunately it doesn't help me.

I can't understand why frequency, wave function, MQ, parallel
transport, etc. can make the down direction prefer over the up
direction.

But thank you all the same for your erudite explanation, a little too
technical for me.

Luigi Fortunati

### Luigi Fortunati

Sep 11, 2022, 7:38:33 PM9/11/22
to
Stefan Ram venerd=EC 09/09/2022 alle ore 08:30:00 ha scritto:
> I currently can't go on websites, but maybe what is intended
> is this: a particle is placed on a geodesic compatible with
> that particle. The particle now can move on the geodesic,
> but how does it know in which direction as the geodesic has
> no preferred direction?
>
> The answer might be: While a geodesic has no preferred
> direction, the time coordinate has.

The time coordinate is part of the geodesic.

If the time coordinate has a preferred direction, the geodesic also has
a preferred direction.

[[Mod. note --
A useful mental model for a geodesic at a point is motion on the surface
of the Earth. That is, starting at some specified point, we can move
in a specified compass direction (e.g., you might start out moving due
west). If, once moving, we don't turn left and you don't turn right,
our motion will be along a geodesic on the Earth's surface.

Thinking of our starting point again, you could have started moving
in any direction (e.g., instead of bearing 090 degrees = due west, we
could have chosen any other compass bearing). So there are a whole
(infinite) family of possible geodesics passing through that starting
point (one for each possible compass bearing).

If I understand you correctly, you're asking "once a particle is moving,
how does it know to continue moving in that direction?". The answer is
basically conservation of momentum: unless there is some external force
pushing on the particle, it's going to continue moving in the *same*
direction it was already moving in.

In terms of geodesics in relativity (the original context of your question),
it's essential to realise that (as others have noted) the trajectoris of
free particles are geodesics in *spacetime*, not geodesics in *space*.
That means the most useful particle velocity to think about is the
4-velocity, which is *never* zero (you're always moving forward in time),
and corresponding momentum is the 4-momentum, which is also never zero.
(Recall that in relativity, even a zero-rest-mass particle like a photon
still has a nonzero momentum so long as it carries nonzero energy.)
-- jt]]

### Luigi Fortunati

Sep 12, 2022, 1:02:09 PM9/12/22
to
Luigi Fortunati domenica 11/09/2022 alle ore 11:38:28 ha scritto:
> The time coordinate is part of the geodesic.
>
> If the time coordinate has a preferred direction, the geodesic also has
> a preferred direction.
>
> [[Mod. note --
> A useful mental model for a geodesic at a point is motion on the surface
> of the Earth. That is, starting at some specified point, we can move
> in a specified compass direction (e.g., you might start out moving due
> west). If, once moving, we don't turn left and you don't turn right,
> our motion will be along a geodesic on the Earth's surface.
>
> Thinking of our starting point again, you could have started moving
> in any direction

Exactly, I could have started moving in any direction and, among all, I
would have been forced, finally, to choose only one.

> in any direction (e.g., instead of bearing 090 degrees = due west, we
> could have chosen any other compass bearing). So there are a whole
> (infinite) family of possible geodesics passing through that starting
> point (one for each possible compass bearing).

That's right, the choice is one of an infinite number of possible
geodesics.

> If I understand you correctly, you're asking "once a particle is moving,
> how does it know to continue moving in that direction?".

No! I am asking how does it know how to "start" to move (along
4-geodesic) and not how to "continue" to move.

Once the particle has moved, its choice (among the infinite possible)

> The answer is basically conservation of momentum

There is no conservation of 4-momentum when the elevator cables break
and the elevator goes from constrained condition to free fall.

> unless there is some external force pushing on the particle, it's going to
> continue moving in the *same* direction it was already moving in.

The elevator where the cables break, does not keep moving in the *same*
4-direction it was moving before.

> In terms of geodesics in relativity (the original context of your question),
> it's essential to realise that (as others have noted) the trajectoris of
> free particles are geodesics in *spacetime*, not geodesics in *space*.
> That means the most useful particle velocity to think about is the
> 4-velocity, which is *never* zero (you're always moving forward in time),
> and corresponding momentum is the 4-momentum, which is also never zero.

When the cables break, the elevator does not retain the 4-momentum it
had before but switches from a certain 4-momentum (the one it had when
standing at the floor) to another completely different 4-momentum (the
one it assumes during free fall).

### Stefan Ram

Sep 12, 2022, 3:30:46 PM9/12/22
to
Luigi Fortunati <fortuna...@gmail.com> writes:
>The time coordinate is part of the geodesic.

When I wrote my last post, I thought of a geodesic as being
a mere subset of space-time, i.e., a set of events like a
line is a set of points, and a mere line has no direction.

But you may be right that it can be seen as having more
structure. Maybe it depends on the details of how a specific
textbook defines it. For the book "Gravitation" by Misner
et al. it's a "curve". I can't find a formal definition for
"curve" in "Gravitation", but it seems to be a continuous
mapping from the real numbers into space-time. So it has a
natural positive direction.

[[Mod. note -- You're right, in relativity a curve is a continuous
mapping from the real numbers (or a closed interval of real numbers)
into spacetime.
-- jt]]

### Richard Livingston

Sep 12, 2022, 3:36:16 PM9/12/22
to
On Sunday, September 11, 2022 at 6:38:33 PM UTC-5, Luigi Fortunati wrote:
> Stefan Ram venerd=EC 09/09/2022 alle ore 08:30:00 ha scritto:
> > I currently can't go on websites, but maybe what is intended
> > is this: a particle is placed on a geodesic compatible with
> > that particle. The particle now can move on the geodesic,
> > but how does it know in which direction as the geodesic has
> > no preferred direction?
> >
> > The answer might be: While a geodesic has no preferred
> > direction, the time coordinate has.
> The time coordinate is part of the geodesic.
>
> If the time coordinate has a preferred direction, the geodesic also has
> a preferred direction.
...

This is somewhat over simplified, but you can think of the g_00 metric component
as similar to a potential, in that it multiplies the particle energy rather than
adds/subtracts like a true potential. Never the less, the "potential" decreases
as you get closer to the gravitating mass, and thus objects feel a "force"
in that direction.

When you really understand how it works, you will find that the curvature
of the metric that causes gravitational acceleration is the time metric,
g_00, that varies with radius. Ultimately this is what causes the acceleration
to be toward the large mass.

The radial metric, g_11, also varies with radius, but this is not an intrinsic
curvature that causes an acceleration. It does affect orbits, particularly
close to the event horizon, and is the reason there are no stable orbits
w/in 1.5 Schwarzshild radii of the event horizon.

Rich L.

[[Mod. note --
For massive particles the innermost stable *circular* orbit in
Schwarzschild is a bit farther out, at r=6M. There are circular orbits
inside r=6M, but they're unstable.

For massless particles (photons et al), there's only one circular orbit,
at r=3M. It's unstable.
-- jt]]

### Jonathan Thornburg [remove color- to reply]

Sep 13, 2022, 2:20:07 AM9/13/22
to
Luigi Fortunati <fortuna...@gmail.com> wrote:
> I am asking how does it know how to "start" to move (along
> 4-geodesic) and not how to "continue" to move.

The simple answer is that it's always been moving. When the particle
first came into existence, it was already moving in spacetime, so it

> There is no conservation of 4-momentum when the elevator cables break
> and the elevator goes from constrained condition to free fall.

In special relativity we start with the notion of a worldline, a particle's
path through spacetime. Introducing $(t,x,y,z)$ coordinates, we can write
this as
t = t(tau)
x = x(tau)
y = y(tau)
z = z(tau)
where tau is the particle's proper time, i.e., the time measured by an
ideal clock carried along with the particle.

The particle's 4-velocity and 4-acceleration are then defined as
$u = (dt/d\tau, dx/d\tau, dy/d\tau, dz/d\tau)$
and
$a = du/d\tau = (d^2t/d\tau^2, d^2x/d\tau^2, d^2y/d\tau^2, d^2z/d\tau^2)$
Finally, we can define the particle's 4-momentum (assuming the particle
to have constant mass $m$) as
$p = mu = (m dt/d\tau, m dx/d\tau, m dy/d\tau, m dz/d\tau)$

[In general relativity things are a bit messier: we define a worldline
in the same way, but 4-velocity and 4-acceleration are now defined in
terms of "covariant derivatives" or "absolute derivatives", which
effectively add some extra terms which depend on the Christoffel symbols.
4-momentum is still given by $p = mu$.]

See
https://en.wikipedia.org/wiki/World_line
https://en.wikipedia.org/wiki/Four-velocity
https://en.wikipedia.org/wiki/Four-acceleration
https://en.wikipedia.org/wiki/Four-momentum
for more on these concepts.

[Note that some authors -- including those of at least the 4-velocity
and 4-acceleration Wikipedia articles -- use a different convention than
I use; they take the time component of 4-vectors to have an extra factor
of c, i.e., they define
$u = (c dt/d\tau, dx/d\tau, dy/d\tau, dz/d\tau)$
$p = (mc dt^2/d\tau^2, m dx^2/d\tau^2, m dy^2/d\tau^2, m dz^2/d\tau^2)$
I'm omitting these extra factors of c.]

When we say that 4-momentum is conserved, that's shorthand for saying
that a particle's 4-momentum is constant *if* there is no net force
acting on the particle.

To discuss your elevator example, we need to decide how we're going to
model gravity, either
(a) Newtonian mechanics or special relativity, in which we model gravity
as a force acting in flat (Minkowski) spacetime, or
(b) general relativity, in which we model gravity as a manifestation
of spacetime curvature.

> There is no conservation of 4-momentum when the elevator cables break
> and the elevator goes from constrained condition to free fall.

The elevator's movement is always governed by the relativistic version
of Newton's 2nd law, which (taking the elevator's mass $m$ to be constant)
is
a = F_net/m
where a is again the elevator's 4-acceleration, and F_net is the net
4-force acting on the elevator.

In perspective (a) [where gravity is a force in flat spacetime]:
--> Before the cable breaks, there are two forces acting on the elevator:
gravity: F=mg downwards
and
cable tension: F=mg upwards
so F_net = 0, the elevator's 4-momentum p is constant, and the elevator's
4-acceleration is 0.
--> After the cable breaks, there is one force acting on the elevator:
gravity: F=mg downwards
so F_net = mg (pointing down) and hence the elevator's 4-momentum p
changes and the elevator's 4-acceleration is nonzero.

In perspective (b) [where gravity is a manifestation of spacetime curvature]:
--> Before the cable breaks, there is one force acting on the elevator:
cable tension: F=mg upwards
so the elevator's 4-acceleration is nonzero (pointing upwards) and
the elevator's path is not a geodesic. This is consistent with the
elevator remaining stationary (i.e.,
x(tau) = constant
y(tau) = constant
z(tau) = constant
is a solution of Newton's 2nd law) because of the extra
Christoffel-symbol terms in the covariant derivatives in the
definition of 4-acceleration.
--> After the cable breaks, there are no forces acting on it, so it moves
along a geodesic, with zero 4-acceleration. That geodesic path starts
(at the moment the cable breaks) with the 4-velocity of the elevator
being at rest (4-velocity having zero spatial components); at later
times the elevator moves downwards.

I wrote [in the context of *general* relativity, where we model gravity
via spacetime curvature, so gravity is *not* a "force"]
>> unless there is some external force pushing on the particle, it's going to
>> continue moving in the *same* direction it was already moving in.

> The elevator where the cables break, does not keep moving in the *same*
> 4-direction it was moving before.

That's right. The elevator's 4-velocity will change after the cable
breaks. In special relativity we say that that change is due to an
external force (gravity) acting on the elevator. In general relativity
we say that that change is due to geodesic motion in a curved spacetime.
It's the same (changing) 4-velocity either way, we're just describing
it differently.

> Ok, let's talk about 4-momentum.
>
> When the cables break, the elevator does not retain the 4-momentum it
> had before but switches from a certain 4-momentum (the one it had when
> standing at the floor) to another completely different 4-momentum (the
> one it assumes during free fall).

Let's work this out in the context of special relativity.
[Doing this in general relativity would be a bit messier.]

Let's take the time the cable breaks to be t=0, and let's take our (x,y,z)
coordinates to be such that the elevator is at rest at x=y=z=0 before
the cable breaks. Finally, let's orient our (x,y,z) coordinates such
that the external gravitational accelration g points in the -z direction.

To simplify the computation, let's assume that the elevator's 3-velocity
is much less than the speed of light. This is fine for investigating what
happens around the time when the cable breaks. This assumption means that
tau = t, so that the time component of 4-velocity is just 1.

Then Newtonian mechancis tells us that the elevator's 3-velocity (the usual
velocity of Newtonian mechanics) is
{ (0 ,0,0) if t <= 0
v(t) = {
{ (-gt,0,0) if t > 0
while the elevator's 4-velocity is
{ (1,0 ,0,0) if t <= 0
u(t) = {
{ (1,-gt,0,0) if t > 0
and hence the elevator's 4-momentum is
{ (m,0 ,0,0) if t <= 0
p(t) = m u(t) = {
{ (m,-mgt,0,0) if t > 0

You can see that v, u, and p are all continuous functions of time.

If we were to analyze the dynamics in general relativity we'd get the same
answers for $v(t)$, $u(t)$, and $p(t)$, but the calculations to get them
would be messier.

### Luigi Fortunati

Sep 13, 2022, 3:03:40 PM9/13/22
to
Jonathan Thornburg [remove color- to reply] marted=EC 13/09/2022 alle ore
08:20:03 ha scritto:
> Luigi Fortunati <fortuna...@gmail.com> wrote:
>> I am asking how does it know how to "start" to move (along
>> 4-geodesic) and not how to "continue" to move.
>
> The simple answer is that it's always been moving. When the particle
> first came into existence, it was already moving in spacetime, so it

Ok, in spacetime the elevator has always been in motion, even when it
is stopped at the floor.

Okay, it's true that his 4-velocity has always been non-zero.

But, when the cables break and the elevator goes into free fall, its
4-velocity changes (i.e. it's a new different 4-velocity that wasn't
there before) or is still the same as it was when it was bound to the
floor?

[[Mod. note -- I think I answered your question in a recent message
which may not have reached you yet:

Let's take the time the cable breaks to be t=0, and let's take our (x,y,z)
coordinates to be such that the elevator is at rest at x=y=z=0 before
the cable breaks. Finally, let's orient our (x,y,z) coordinates such
that the external gravitational accelration g points in the -z direction.

To simplify the computation, let's assume that the elevator's 3-velocity
is much less than the speed of light. This is fine for investigating what
happens around the time when the cable breaks. This assumption means that
tau = t, so that the time component of 4-velocity is just 1.

Then Newtonian mechancis tells us that the elevator's 3-velocity (the usual
velocity of Newtonian mechanics) is
{ (0 ,0,0) if t <= 0
v(t) = {
{ (-gt,0,0) if t > 0
while the elevator's 4-velocity is
{ (1,0 ,0,0) if t <= 0
u(t) = {
{ (1,-gt,0,0) if t > 0
and hence the elevator's 4-momentum is
{ (m,0 ,0,0) if t <= 0
p(t) = m u(t) = {
{ (m,-mgt,0,0) if t > 0

You can see that v, u, and p are all continuous functions of time.
-- jt]]

### Luigi Fortunati

Sep 15, 2022, 7:12:42 PM9/15/22
to
Richard Livingston luned=EC 12/09/2022 alle ore 07:36:11 ha scritto:
>> The time coordinate is part of the geodesic.
>>
>> If the time coordinate has a preferred direction, the geodesic also has
>> a preferred direction.
> ...
>
> This is somewhat over simplified, but you can think of the g_00 metric
> component as similar to a potential, in that it multiplies the particle
> energy rather than adds/subtracts like a true potential. Never the less, the
> "potential" decreases as you get closer to the gravitating mass, and thus
> objects feel a "force" in that direction.

Why is "force" in quotes?

If you feel it, it is a force without quotes.

[[Mod. note --
In the context of general relativity (GR) you don't feel gravity as
a force.

In GR the apparent downward "force" which I feel right now (sitting in
a chair in a room near the Earth's surface) is understood not as a
gravitational "force", but rather as follows:
(a) An inertial reference frame (IRF) at my location is by
definition in free-fall, and therefore must be accelerating
downwards at 9.8 m/s^2 with respect to the Earth's surface.
(b) Therefore, my chair (which is stationary with respect to the Earth's
surface) is accelerating *upwards* at 9.8 m/s^2 with respect to
an IRF at my location.
(c) Because my chair is accelerating upwards at 9.8 m/s^2 with
respect to an IRF, my chair must push up on my body in order
to accelerate my body upwards at 9.8 m/s^2 with respect to an IRF.
(d) More generally, because a coordinate system fixed to the Earth's
surface is accelerating upwards at 9.8 m/s^2 with respect to an
IRF, every object feels an apparent "force" pointing downwards.
-- jt]]

### Rock Brentwood

Oct 7, 2022, 4:16:58 AM10/7/22
to
On Wednesday, September 7, 2022 at 1:22:46 AM UTC-5, Luigi Fortunati wrote:
In any metric geometry, around each point is a neighborhood U,
such that between any two points A and B in U
lies a unique geodesic between A and B.
Uniqueness means it's therefore the same regardless of direction.

This even includes non-Riemannian geometries,
like the chronogeometry of Newtonian gravity,
which can be regarded as residing on a light cone
of the 4+1 dimensional pseudo-Riemannian chronogeometry
given by the metric whose line element is

dx^2 + dy^2 + dz^2 + 2 dt du - 2V dt^2

where V = -GM/r (with r^2 = x^2 + y^2 + z^2)
is the gravitational potential per unit mass.

This also works perfectly well for multi-body potentials,
V = -sum_a (-GM_a/|r - r_a|).

Its geodesics comprise the orbital motions of Newtonian gravity
AND the geometrical instantaneous geodesics of space at any instant.

You may be hard-pressed, in this case, to account for
what the corresponding geodesic *distances* are,
since everything is zero on a light cone!
All the geodesics for Newtonian gravity
are null curves in 4+1 dimensions.
But it's the same in both directions.

This may be compared to the Schwarzschild metric
which is equivalently given as light cones
in the 4+1 dimensional chronogeometry
whose metric has the following line element

dx^2 + dy^2 + dz^2 + 2 dt du - 2V dt^2
+ alpha du^2 - 2 alpha V dr^2/(1 + 2 alpha V)

where dr^2 = (x dx + y dy + z dz)^2/r^2 and alpha = (1/c)^2.

This is a deformation of the Newtonian chronogeometry,
from alpha = 0 to alpha > 0.
Here, however, V = -GM/r is only for one body, not many.
There may exist similar deformations for the many-body cases.

The geodesic distances on the light cones
in the (4+1)-dimensional chronogeometry are all 0,
since all the curves on the light cones are null.
The light cones, however (unlike the Newtonian case)
comprise metric geometries in their own right -
that is: the Schwarzschild chronogeometries,
whose the metric yields - for time-like curves -
a proper time equal to s = t + alpha u.
It's the same in both directions.

In the alpha = 0 case, which is Newtonian gravity,
the (3+1)-dimensional chronogeometry continues to have
s = t + alpha u as an invariant ... which is now just s = t.
So, it can still be taken as a metric of sorts for time-like curves.
Ordinary time is proper time for Newtonian gravity.