Re: What is the nature of gravitational curvature

Dear Les,

This is great question and leads to some fundamental ideas about space-
time.

What do we mean by saying that space is curved?  Or, even more simply, 
what do we mean by saying that a surface is curved?  If we were living on 
a curved surface, how would we know that it was curved?  Well, the fact of 
the matter is, we are living on a curved surface, the Earth's surface, and 
it is certainly not flat.  How could we prove this?  Imagine starting a 
journey at the North Pole.  We travel south until we reach the equator, 
and then turn 90 degrees to the left.  We move along the equator one-
fourth of the way around the Earth and then turn 90 degrees to the left 
again, and go north until we end up back at the North Pole.  Our journey 
has consisted of three straight sections connected with three 90-degree 
turns.  If we were to try and draw our trip on a flat piece of paper it 
would be a triangle with three ninety-degree angles!  That contradicts 
what we know about Euclidean geometry; you can't have a flat triangle with 
three ninety-degree angles.  That proves that the Earth is not flat.

Now, how can we apply this to space?  If we could take an analogous trip 
as described for the Earth, and if it didn't match what we would expect 
for a flat Euclidean three-dimensional space, then we would have to assume 
that space was not flat.  Rather than making a big triangle, another 
possibility is to travel in a big circle, with the center of the circle at 
some massive object, like the Sun.  It turns out that the distance around 
our circle, the circumference, is not equal to 2*PI times the radius of the 
circle!  Again, this contradicts what we know about flat Euclidean space, 
so our space must be curved!

A very important point about this space curvature can be seen when we 
compare the curvature of space with the curvature of a piece of paper.  To 
curve a two-dimensional flat sheet, we usually think of it as bending into 
the third dimension.  With space, we are really talking about a four-
dimensional space-time, in which we combine the three spatial dimensions 
with the dimension of time.  To curve space-time, we do not need a fifth 
dimension; space-time can actually curve in itself.

Now, what about moving objects?  You are correct in that objects follow 
straight lines through curved space-time.  You can think of a 'straight' 
line in space-time as being the shortest line connecting two points.  
However, straight lines do not depend only on the space part of the 
motion, but also on the time part.  A straight line that moves mostly in 
the time direction (that is, one with little, or slow, space-like motion) 
will be very different from a straight line that has equal space and time 
motions (as for a ray of light), for example.  And so, objects moving at 
different speeds will travel along different paths, since the space-part 
of their straight lines, through curved space-time, will be different.

If you would like to do some more reading about this, I think a great book 
that discusses the experimental verifications of Einstein's theories 
is "Was Einstein Right?: Putting General Relativity to the Test" by 
Clifford M. Will.

I hope that answers your question.  Please let us know if you would like 
any more information.

Thank you for your interest.

Sincerely,

Jim Guinn
Georgia Perimeter College