Sciences
The
solution to Einstein's equations for a threedimensional spacetime with
a
single point mass looks like a flat spacetime everywhere to local
observers 
except when they make measurements around spatially closed curves
(curves that
come back to the same place, but at a later time) that contain the mass
itself.
When they do this, they learn that the space part of the spacetime they
live in
is not a flat plane, but a cone.
A
Flatlander travelling around the blue arc will think she's been around
a complete
circle. But she will measure the circumference of this
circle to be smaller
than 2 R Pi, or 6 Pi in this case. The angular size of the missing
wedge above
is Pi/2, so the Flatlander will measure the circumference of her path
to be 3 R
Pi/2, or 4.5 Pi. Therefore she will be able to deduce that her path has
encircled a point mass, and that she must live on a cone, not on a flat
plane.
Note that circular paths that do not circle the mass will have the
normal
circumference of 2 R Pi
The missing angle is called the deficit angle.
According to Einstein's
equations, the mass M of the point is related to the deficit angle B
through the
formula B=8GM Pi, where G is Newton's constant. The deficit angle can't
be
larger than 2Pi, so there is a limit on the allowed mass: M must be
less than or
equal to 1/4G. (Note: the units of Newton's constant are not the same
in three
and four spacetime dimensions. Why not?)
Another way of picturing this cone is shown below. It's just the xy
plane with a
wedge removed, and the sides sewn back together. Note that the blue
circular arc
is actually a closed path when the red dashed lines are identified by
the
sewingtogether operation.
As we've gone over before in previous sections, in one time and two space dimensions, if we use the Einstein equations to obtain the spacetime metric for the spacetime around one massive point particle, all the curvature is concentrated where the mass is and the spacetime around the point mass is flat but with a deficit angle, as depicted in the figure below. We see light flashes 1 and 2 leave from a common point at the same time until they cross where we've put the deficit angle gap. This crossing looks strange in flat coordinates, it looks like the flashes bounce from the gap and change identities. 
If we get rid of the deficit
angle by transforming to another set of coordinates
where the angle is rescaled by that amount, we get a smooth picture
that doesn't have to be sewn together across some
angular gap, but the null geodesics in these new coordinates
no longer look like straight lines, as shown below:

If we look
at the paths we've seen in the last two sections in spacetime,
where time has its own coordinate axis, the picture
for low mass expands to what we see below. Notice that the places where null geodesics cross are marked as points conjugate to the event t=0, x=1, y=0 through which they all pass initially. This behavior of null geodesics reconverging or crossing after emanating from a common source is of crucial importance to all the odd and interesting behaviors that can happen in the Genreal Relativity model of Nature. Black holes, naked singularities, clothed singularities, wormholes, time travel  they all involve either an encouragement or a frustration of this nullgeodesiccrossing phenomenon.

Here is
the light cone of our event in question, which is
the surface made by all of the paths of light or null
geodesics as we call them, that pass through the
spacetime event t=0, x=1, y=0. Now let's compare this with the previous of the light cone in flat spacetime. (Bottom Picture.) Notice that the light cone below has big extra flap hanging on the inside. This extra flap is made of all the null geodesics that crossed a second time after leaving the event t=0, x=1, y=0, and then veered away from the main body of the light cone after that second crossing. Running down the middle of the extra flap, where the flap meets the normal part of the light cone, is the line of points conjugate to the original crossing event t=0, x=1, y=0.

light cone in flat spacetime.

Causal structure of this spacetimeThe deflection of light by gravity (such as it happens in General Relativity) does not blur at all the crisp and wide boundary between the past and future shown in the figure below. We can see quite well below what region of this spacetime is contained within the boundary of the causal future of the event E, what region of this spacetime is contained within the boundary of the causal past of the event E and what regions are outside of both the past and the future of this event. 