Re: Does gravity really curve space?

Dear Chris,

I’m afraid I must disagree with the author of the article you sited.  I 
don’t agree that space-time and curved space-time are purely “mathematical 
constructs”, rather they seem to be the most natural way of describing the 
phenomena we see around us, like gravity.  I wouldn’t say that they are 
the only way of describing gravity, but we seem to be led to them by the 
simplest assumptions that we can make.

Before we talk about curves, let’s think about straight.  What is a 
straight line, and how might we define it?  Keep in mind that in general 
we must consider points that are not only at different places in space, 
but at different times, too!  But let’s start simple.  If we have two 
points in space at the same time, what is a straight line that connects 
them?  Well, why not the line with the smallest distance, such as that 
formed by a “taut rope” as in the article.  That certainly seems 
reasonable, and we can accept that for the time being.  Now what if we 
consider two points (typically they are called “events”) that are at 
different places in space and at different times, too.  What is a straight 
line that connects these two events?  You can’t stretch a rope between two 
different times!  Let’s pick two events within a coordinate system, for 
example the event at the origin (t, x, y, z) = (0, 0, 0, 0) and the event 
10 second later and 10 meters along the x-axis, or (t, x, y, z) = (10s, 
10m, 0, 0).  What is the straight line that connects these two?  You might 
guess the line that changes its “t” and “x” values in a uniform way, as 
(0, 0, 0, 0), (1s, 1m, 0, 0), (2s, 2m, 0, 0), (3s, 3m, 0, 0), (4s, 4m, 0, 
0), (5s, 5m, 0, 0), (6s, 6m, 0, 0), (7s, 7m, 0, 0), (8s, 8m, 0, 0), (9s, 
9m, 0, 0), (10s, 10m, 0, 0), might seem reasonable.  What is this line?  
You might say it is the line taken by a body moving through space and time 
with a constant speed, 1m/s in the positive x direction.  That is, the 
straight line corresponds to that taken by a body moving with no 
acceleration between the initial and final events.  (Keep in mind that 
this idea would not work if the events are too far apart for a given time, 
this is because a body would have to travel faster than the speed of light 
to get from one to the other.)  A general statement would then be that 
straight lines in space-time correspond to the motion taken by a freely 
moving body between the initial and final events.  Does that sound 
acceptable?  I hope so.  We call these lines “geodesics”.  Now, what is 
the “distance” between these two events?  That’s another problem.  The 
spatial concept of distance will not work, because if we look at these two 
events from a different reference frame, the measured distance will be 
different.  For example, the body that moves between them will be at each 
event, and so thinks that the events occur at the same spatial position.  
The moving object will measure the spatial separation of the events to be 
zero!  This means that ordinary distance is not very useful when dealing 
with events since its value depends on who is measuring it.  I won’t prove 
this (you can find it in just about any introductory text on special 
relativity) but a useful invariant “distance” (it doesn’t change with 
different reference frames) is the “interval”.  As defined in your cited 
article, in flat space-time the interval is defined as ds^2 = -c^2 dt^2 + 
dx^2 + dy^2 + dz^2 (it can be defined like this or with a relative minus 
sign, it doesn’t really matter).  The presence of the “c” just converts 
the units of time to that of space, that is, seconds to meters.  Typically 
we can measure time and space with the same units by setting “c=1” so that 
1 second = 3x10^8m.  This is nothing more that unit conversion.

Now, back to the article.  The author claims that by manipulating the 
variables in the interval, the “purely time-like” value is made to 
look “length-like” by multiplying by c.  I’m afraid this doesn’t make 
sense.  An interval is either time-like or space-like depending on whether 
it is positive or negative.  Given our definition, ds2 = -c^2 dt^2 + dx^2 
+ dy^2 + dz2, a positive interval is space-like, a negative interval is 
time-like, a zero interval is called “null” or “light-like”.  You cannot 
make an interval “look” like something else by multiplying it by c.

The author also suggests stretching a rope between two points on the 
Earth’s orbit and states that this distance is less than that moved by the 
Earth in the orbit, and therefore geodesics do not minimize distances.  
Geodesics do not minimize distances, they minimize intervals from event to 
event along an objects motion.  The “rope distance” from A to B in the 
article is certainly less than the spatial distance the Earth travels, but 
the Earth also has a coordinate time change of three months, or so, which 
is not taken into account in the “rope distance”.  As we mentioned before, 
the rope is not stretched between two events that are three months apart, 
and so cannot be used to compare the interval moved through by the Earth.

So is space-time curved?  Well, what is flat space-time?  One in which 
parallel lines never intersect; one in which coordinates can be defined in 
a Euclidean fashion and the interval always has the form ds2 = -c^2 dt^2 + 
dx^2 + dy^2 + dz2.  Unfortunately, or fortunately if you like this sort of 
thing, in the presence of gravitating bodies, initially parallel lines can 
intersect and there is no way to have an invariant interval with the flat 
space-time form.  It turns out that it is even impossible to have a series 
of clocks that can be maintained to run at the same rate.  Therefore, we 
seem to be led to the idea of curved space-time.  Using whatever 
reasonable way we want to define straight, space-time doesn't seem to be 
flat!

There are many great books that introduce Special and General Relativity.  
For General Relativity I would suggest “A First Course in General 
Relativity” by Bernard F. Schutz.

Well, I hope I have answered your question, Chris.  If you would like some 
more information, please let us know.

Sincerely,

Jim Guinn