How are geodesics different from gravitational field lines?
Short answer: They're verrrry different.
Longer answer:
1) Geodesics are by far the simpler and more fundamental concept.
There was a decent understanding of straight lines (e.g. Greek
geometry) a couple thousand years before there was any serious
understanding of gravity (Galileo and Newton).
On the surface of the earth, or the surface of a basketball,
the geodesics are the great-circle routes, for reasons having
nothing to do with gravitation.
Geodesics are exceedingly fundamental. In the first paragraph
of the introduction to the Principia, Isaac Newton said:
“The description of right lines and circles, upon which
geometry is founded, belongs to mechanics. Geometry does
not teach us to draw these lines, but requires them to
be drawn”.
You can't even express the first law of motion unless you
have a practical way to draw straight lines in spacetime.
2) We can use geodesics to quantify the effect of gravity.
Gravitational tidal stresses cause geodesics that started
out parallel to become non-parallel.
However, even then the connection between "field lines" and
gravitation is indirect. It's messy and tricky, in part
because the word "gravity" is commonly used to describe two
different things:
2a) There is the local gravitational field g in some chosen
reference frame. This is frame-dependent. To leading
order you can make it vanish, locally, by choosing a
different frame, i.e. a local freely-falling frame.
This g is a vector field. It can be represented, to a
fair approximation, by field lines.
A uniform g field does *not* contribute to geodesic
deviation ... which should be obvious from the fact that
geodesics have physical reality, independent of what
reference frame (if any) you choose, whereas g is
grossly frame-dependent.
2b) There is also the concept of gravity as embodied in
Newton's law of universal gravitation (or, if you want
to get fancy, general relativity). This is completely
frame independent. It describes how the the gravity
on one side of the earth /differs/ from the other side.
In other words, tidal stress. You can't make that go
away by changing the reference frame. This is what
contributes to geodesic deviation.
We are talking about a tensor field. Generally speaking,
representing tensor fields is a royal pain. You can get
some idea by looking at how the force-vectors /change/
as a function of position.
I find it really annoying that typical textbooks switch
back and forth between the frame-dependent and frame-
independent notions of gravity without warning. It's
no wonder students are mystified.
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Here's another way to organize the ideas: Under mild
assumptions, we can expand the gravitational potential
in a Taylor series as a function of position:
-- The zeroth-order term is a scalar field. It is
constant throughout space. This term has no relevance
to geodesics or any other physics, and you can make it
go away by fiat, i.e. by a gauge transformation.
-- The first-order term is a vector field. This is the
familiar g field. It can be represented by field lines.
You can make this Taylor term go away completely by
choosing a freely-falling reference frame. Geodesics
don't care about this term.
-- The second-order term is a tensor field. This is
the tidal stress. It is frame-independent. You can't
make it go away. This drives geodesic deviation.
You can make paper models of a gravitational potential,
showing how the g field is different from the tidal stress.