REACHING INFINITY IN A RAY OF LIGHT

"The infinity of the Universe is a great blessing for human beings, because human beings' desire for discovery is also infinite!" Mehmet Murat Ildan.

The concept of infinity has fascinated humanity since the early Greek mathematicians began to unravel its secrets. Archimedes was able to calculate the surface area and volume of a wide variety of curved bodies and realized that this value was "hidden" in infinity.

In this article, we'll discover a strange and fascinating relationship between the speed of light, infinity, and certain very peculiar space-times. This relationship makes it possible for a ray of light to reach infinity and return to its starting point in a finite time.

Next we will enter a strange place where infinity is not a mathematical entity but a physical place that can be reached through a ray

light.

The strange AdS space-time

The first cosmological solutions to the equations of general relativity offered two possible universes: one with curvature and a positive cosmological constant, and another with curvature and a negative cosmological constant. In honor of their discoverer, Willem de Sitter, the former were called de-Sitter (dS) spacetimes, and the latter anti-de-Sitter (AdS) spacetimes. In this article, we will analyze the incredible properties of the latter.

Visualizing curved spaces can be very complicated. A space with positive curvature, like a sphere, closes in on itself and therefore has a finite volume. However, a space with negative curvature continually opens outward, and therefore, this space has no edges and infinite volume:

If we add time to a three-dimensional AdS space, then we have a four-dimensional AdS space-time whose metric can be written as:

The constant "a" measures the radius of curvature, and "omega" represents the metric of a 2-sphere of unit radius. An AdS spacetime represents a Universe with constant negative curvature. Since gravity is actually the curvature of spacetime, any observer within an AdS geometry will be immersed in a gravitational potential.

To analyze the characteristics of this geometry, we must analyze the trajectory a body takes in this space-time when it is freely released; that is, we must visualize the geodesics. To do this, we make the following coordinate change in the previous metric:

Then we get the metric:

Considering a body at rest that moves only in the radial coordinate we obtain:

The geodesic is given by:

For "timelike" journeys, that is, for travelers with speed less than c, we have:

This is the equation for a simple harmonic oscillator with period T=a. This is really quite strange: A body in AdS geometry follows cyclical trajectories in spacetime! This means that time is cyclical: after a period of time 2πa has elapsed, any observer starting from a point P will return to the same starting point!

To understand how strange this is, consider an astronaut in an AdS spacetime. The astronaut throws a ball upward with a velocity V1. The ball will rise, stop, and fall back toward the astronaut in a time 2πa. The astronaut then launches the ball with a catapult that propels it with a velocity V2 greater than the previous one. The astronaut, astonished, observes that the ball will return in the same time as before! The difference is that the ball has traveled a greater distance than in the previous case.

After observing this, the perplexed astronaut uses a very powerful cannon to propel the ball. The ball acquires an initial velocity V3 much greater than the previous ones, however, after the same time 2πa has elapsed, the ball returns to its starting point! This strange Universe is determined by cycles of constant time!

As the ball's initial velocity increases, the distance traveled increases, and therefore the elliptical trajectory described by the ball becomes more and more eccentric. However, the time it takes to fall is always the same: 2πa

This strange behavior is best understood by observing that in AdS geometry time is "wound" around the hyperboloid forming circles of length 2πa (1):

The key question we ask next is: What will happen if we use a laser beam that escapes from the observer at the maximum permissible speed c? The answer to this question is very hard to believe: light will travel an infinite distance in a finite time! How is this possible?

Reaching infinity in finite time

How is it possible to travel an infinite distance in a finite time? To try to understand this as intuitively as possible, we'll use a geometric image. The first step is to draw a "three-dimensional slice" of AdS spacetime. This will be the hyperboloid with the equation:

Where L is a positive real number.

A hyperboloid is the figure obtained by rotating a hyperbola around its vertical axis. The equation of a hyperbola is:

Where a and b are the "vertices" on the x,y axes:

The dotted diagonal lines are the asymptotes of the hyperbola and are the lines with slope b/a. As we increase the value of the X axis, the hyperbola tends asymptotically to the line Y=b/aX so that this could only be reached by the hyperbola at infinity.

Let's return to our three-dimensional hyperboloid. The intersection of the hyperboloid with the plane bounded by the XY axes is a circle (this can be easily seen by setting Z=0 in the hyperboloid equation), so L will be the radius of that circle:

The metric of the hyperboloid can be written as ds2=dx2+dy2−dz2 . If we consider a body moving radially along the X-axis (hence dy2=0) we have: ds2= dx2-dz2. This is the equation of a hyperbola with a=b=1:

The asymptote of the hyperbola is the line with unit slope y=x.

By special relativity we know that if the metric of a path is positive (2), that is, "timelike", the body moves at speeds less than c and if the metric is zero the path is that of a light ray traveling at speed c. The case of a body at rest corresponds to the largest positive value of the metric, that is, when dz2=0. Therefore, an observer at rest will move in the plane Z=0. An observer at rest at time to at point Po will take the following trajectory in AdS spacetime:

In general relativity, an observer's proper time is given by the length of the path (the world line) they make through four-dimensional space-time. In this specific case, for an observer at rest, this path is simply 2πa.

For a moving body our metric ds2=dx2-dz2 will now have two components: one on the X axis and one on the Z axis, so the trajectories will be on a plane inclined with respect to the XZ axes. This plane is therefore the ax+bz plane:

The slope of the plane will be b/a. The intersection of this plane and the hyperboloid is an ellipse. The ellipse cuts the circle at two points: Po and P1. The semicircle between Po and P1 in the plane Z=0 is the path taken by the observer at rest and the semiellipse of the inclined plane is the path taken by the ball with speed v. As v increases the plane inclines more and more (the quotient b/a determines the slope of the plane, if a>b the metric is positive but as b and a become equal the ellipse becomes longer and longer and the trajectory approaches that of a light ray (ds2=0 and 45º inclination).

Now we come to the crucial point: when we reach the limiting slope of 45°, we obtain the path of a light ray, and this path coincides with the asymptote of the hyperboloid. This means that the path taken by a light ray ends at the asymptotic edge of AdS spacetime, which is located at an infinite distance from any other point in the AdS geometry.

However, as can be seen in the figure, point Po represents the moment at which the object is thrown by the observer and P1 the moment at which the object returns to the observer. As can be seen , regardless of the inclination of the plane, the observer at rest (red line) measures a finite time 2πa!

The key fact is that independently of a and b, i.e. the initial energy, the ellipse cuts the circle in 2 points whose arc is finite and therefore the proper time measured by the observer is finite even in the case where the path is an ellipse of infinite length!

In an AdS spacetime, infinity can only be reached using a light ray. In fact, if we place a mirror at the infinity edge, the ray will bounce off the mirror and reach the starting point in a finite time from the reference frame of an observer at rest. In an AdS spacetime, infinity is a physically localized place! It should be noted that for a light ray, proper time is not defined (there is no reference frame in which light is at rest). This, combined with the strange AdS geometry, allows us to "witness" one of the strangest phenomena imaginable.

AdS space-time is certainly strange and exotic, however, there is one place in our Universe where this geometry could actually exist (2): inside black holes.

As if this weren't astonishing enough, the infinite edge of AdS geometry hides a profound secret related to the nature of spacetime: all information about the four-dimensional AdS geometry can be holographically encoded within it. The famous AdS/CFT conjecture tells us that a three-dimensional quantum system with conformal symmetry formulated at the edge of an AdS spacetime is equivalent to the four-dimensional AdS spacetime itself. This is possible because both systems have exactly the same set of symmetries and are therefore mathematically equivalent.

Using conformal coordinates the AdS hyperbolic geometry can be mapped onto a cylindrical geometry where the infinite edge is translated to the edge of the cylinder.

For all this we can affirm that infinity holds great secrets, including one that could resolve the ultimate nature of space-time.

"If the doors of the perception, everything would appear to man as it is, infinite." William Blake.

Grades:

( 1) In reality, the time coordinate does not "wrap" around the AdS geometry forming closed circles, but rather forms a solenoid to avoid closed time trajectories that violate causality. The period of the solenoid is 2πa.

(2) Depending on the convention the "timelike" metric can be positive or negative depending on whether the time coordinate is considered negative or positive.

(3) The geometry of certain black holes can be considered AdS only under certain specific conditions.

Sources:

The bizarre anti deSitter spacetime , Geometry of (anti) deSitter spacetime , Physics stack exchange