The Horizon Problem for Faster than Light Travel

Hello all,

As I have mentioned in previous posts, a major hurdle for autonomous FLT travel is the horizon problem. But what is a horizon? Is it some single well-defined concept or a catch-all term for many phenomena? In this post, I present some thoughts on horizons in general relativity theory and their implications for FTL travel using warp drives.

A horizon is a limit to the region that can be observed or communicated with in a space. On Earth, horizons can be used to refer to edge of what can be seen when looking out over a vast body of water like an ocean. The limit to what can be seen in this case is caused by the curvature of the Earth's surface and is a cute problem to calculate how distant the horizon appears given the height of an observer above the surface of a spherical Earth (ignoring effects of the Earth's atmosphere) that is straightforward to solve as it is embedded in a Euclidean geometry. 

An observer looking out to the Earth's horizon from a height h above the surface.

In relativity theory, a horizon is also a boundary on the region which an observer can see. Horizons pose an issue for superluminal warp drives as all known warp bubbles create horizons at superluminal speeds, implying that one cannot control the bubble to accelerate or decelerate it or pass communications between observers inside and outside the bubble once it is traveling faster than light. This is the essence of the horizon problem for FTL travel. There are several different types of horizons that are commonly found in GR space-times. Here are several and some of their implications:

Light Cones:

This is not considered a true horizon, but I am mentioning it to better distinguish true horizons. A past light cone indicates the region of space-time an observer can see at one moment, as opposed to a more strict horizon which determines what all observers in a region of space-time can ever see. Light cones about a space-time event are defined as the points in space-time connected to that event by incoming light rays. The cone term comes from the shape of the region close to the point of observation when considering a flat 3 dimensional space-time (see figure). In 4 dimensional space-time, the cone takes the form of a sphere. Light ray trajectories are given by light-like, or null, vectors, the name given to non-zero vectors X that have zero (null) inner product via the space-time metric (g(X,X) = 0).

Stib at en.wikipedia, CC BY-SA 3.0 , via Wikimedia Commons

A light cone varies from moment to moment for a time-like observer such as a person. It is not meaningful on its own for differentiating what is contactable from what is not for observers moving with constant velocity. Case in point: a conversation using light beams can occur between any two time-like inertial observers in a flat space-time, it just might take a while.

The story is more complex for curved space-times, which is where we will find true horizons.

Killing Horizon:

Our first type of horizon is the Killing horizon. A Killing horizon (named after Wilhelm Killing) is defined as a null hypersurface (space of lesser dimension than the full space-time where all tangent vectors are light-like) given by the vanishing of the (metric) norm of a Killing vector field (also named after W. Killing). This definition appears similar to the light cone definition above extended for curved space-times, but is actually much more broadly useful as we will see. To understand this, we must learn what a Killing vector field (KVF) is.

A KVF is a smooth vector field, such as V, that preserves the space-time metric, meaning that the metric is unchanged when shifted in the direction of V. KVFs identify symmetries in a geometry, such as rotational symmetry and time invariance.  

So, a Killing horizon is produced over regions where a KVF's norm vanishes (g(V,V) = 0). Killing horizons can identify when the vectors generating a symmetry of space-time  change signature, transitioning from being space-like (g(V,V) > 0, using the "mostly plus" metric sign convention) to time-like (g(V,V) < 0) or vice versa.

For example, the space-time of a static rotating black hole has a symmetry that looks like time translation far away from the black hole and is generated by a KVF that changes from time-like to space-like in signature as it crosses a horizon called the ergosphere. Observers that cross this horizon will also change signature from time-like to space-like. I will talk more about the ergosphere Killing horizon and other black hole horizons below.

Black Hole Horizons:

The event horizon around spherical Schwarzschild Black Holes is likely the most popularly known horizon in GR. Less widely known is that Black Holes (BHs) can have up to four unique horizons when considering their spin and electric charge. For simplicity, we will limit ourselves to static BHs and their horizons. Dynamic horizons are of interest when considering the lifecycle of a BH, and a warp drive, and I may delve into them more in a future post. I touch on them briefly below in the context of cosmological horizons.

Cartoon of charges spinning BH structure, including some of the horizons (Original image found in Tito and Pavlov 2018)

The horizons are determined where the radial or time components of the space-time metric (grr and gtt) invert in sign (passing through zero). After a convenient choice of units, finding the zero points to the grr and gtt metric components is equivalent to solving the respective equations:

r² + e² + a² - 2 M*r = 0

r² + e² + a² Cos² θ - 2 M*r = 0

where M represents the total mass of the BH, e the total electric charge, and a is a parameter representing the spin of the BH. Solving for the radial coordinate r shows that horizon surfaces occur at

r ^±_ergo = M ± (M² - e² - a² Cos² θ)^1/2

r ^±_event = M ± (M² - e² - a²)^1/2

where the "±" indicates a choice in whether to use a plus or a minus. There are up to four unique solutions and horizons to this space-time, depending on the choice of spin and charge parameters. These four horizons also correspond to Killing horizons for KVFs which point in the time and radial directions far away from the BH, in the BH's rest frame of reference. The horizons at which the time-like KVF changes sign are known as the ergospheres, and the horizons at which the radial KVF changes sign are the event horizons. It is also worth noting that the singularity inside the BH is not point like in general, but shaped like a ring.

The outermost horizon is called the (outer) ergosphere, also known as the stationary limit. The ergosphere shape is actually not spherical in general but oblate. Again, the outer ergosphere coincides with a time Killing horizon caused by the metric time component switching signs (negative to positive in our convention), and for a spinning BH is in part influenced by what is called "frame dragging", a geometric effect induced by the BH spin. For a spinning BH with zero electrical charge, the outer edge of the ergosphere signifies where the frame dragging effect exceeds the speed of light. Observers inside the ergosphere must be moving faster than light as measured by observers outside of the BH. Frame dragging is analogous to the effect used to create warp drives space-times. Stationary objects inside the ergosphere will fall further into the BH, but objects co-rotating quickly enough can maintain distance or even escape if they are imparted some thrust. As some objects can escape, the outer ergosphere does not qualify as an event horizon. An event horizon being defined as a boundary beyond which events cannot affect an observer.

The next horizon is the outer event horizon. This horizon coincides with the outer radial Killing horizon caused by the inversion of the metric's radial component, and implies that between the outer and inner horizons the metric has changed the signature of two of its four principle directions. Objects that are time-like far from the BH and fall past the event horizon have no known means of escape in classical physics. 

The two innermost horizons sit immediately outside of the BH singularity. The inner horizons emulate the outer ergosphere and event horizon shape, but their behavior is opposite: inside the inner ergosphere the metric time component again becomes negative and the time KVF again becomes time-like;  the radial metric component again becomes positive and the radial KVF again becomes space-like. The metric principle directions have regained the signature they had outside the BH.

Cosmological Horizons:

The last example here is from cosmology, and pulls together several of the principles we covered above. Models of big bang cosmology include multiple observational horizons set by the physics and history of our expanding universe. The context for this example with be FLRW (Friedmann–Lemaître–Robertson–Walker) model of cosmology where the stress-energy sources are homogeneous (take the same value everywhere at a given time) and isotropic (have no preferred direction), and for our purposes flat (the space portion of the space-time geometry is neither positively curved like a sphere nor negatively curved like a hyperbolic surface, but flat like Eulidean space). The only remaining geometric degree remaining is the time-dependent scale factor of the universe, a(t).

The near uniform expansion of our universe has served as inspiration for faster than light travel research due to the seemingly cumulative nature of separation speeds as objects become increasingly separated. As an analogy, think of points on a balloon separating as the balloon inflates: the points close together separate less rapidly than points with larger distance between them.

Boy blowing up balloon to demonstrate cosmological expansion. Image from Fun with Astronomy

We are currently only able to see objects up to about 46.5 Gly (giga-light-years) away from the Earth, limited by a wall of primordial radiation called the cosmic microwave background (CMB) emitted from a cooling atomic plasma nearly 13.8 Gyr  (giga-years) ago, when our universe was an estimated 380,000 years old. We cannot see further back using conventional telescopes as the opacity (optical scattering dense-ness) at times before the CMB is too large for light to have reached us undisturbed. This also known as the surface of last scattering for photons. There are also surfaces of last scattering for other types of radiation that reach beyond the CMB, such as neutrinos and gravitational waves. Though not true horizons, these surfaces are practical horizons as it is effectively impossible to perform observations beyond them using their radiation of choice.

Cosmologies also contain a general particle horizon, the farthest distance to emissions traveling at light speed could travel to the observer over the history of the universe. Its reach is limited by the lifetime of the universe, and the objects that can or cannot be seen by a particular observer are dependent on the location of the observer, so it is an observer-dependent horizon. Also, the edge of this observable region is not calculated simply by multiplying the age of the universe by the speed of light, otherwise we would only be able to observe to a distance of about 13.8 Gly, with the history of expansion making the particle horizon distance much larger.

Cosmologies can also contain event horizons. An event horizon to an observer is the distance beyond which we will never see light from. This horizon only exists if the cosmology expands eternally without slowing, as otherwise the horizon would shrink and one would eventually be able to see light emitted from objects at any distance. Note that like the particle horizon, the location of this horizon is still dependent on the location of the observer.

The last horizon type discussed here is the Hubble sphere (named after Edwin Hubble). The radius of the Hubble sphere is set by the distance at which co-moving objects are receding from us at light speed, meaning everything inside the sphere is co-moving away from us at less than the speed of light and everything outside the sphere is co-moving away faster than the speed of light.

Figures showing the relative locations of these horizons for an example cosmology with 30% dark matter and 70% dark energy is below.

Figures of evolving horizons for a cosmology with 30% dark matter and 70% dark energy. The original annotated version can be found on Dr. Tamara Davis's website here.

You may have noticed that the event and Hubble horizons shown are no longer growing in proper distance/size (as measured by a fixed ruler) and are shrinking in co-moving size (as measured by rulers that scale with the cosmology's expansion factor a(t)). This is a feature of the cosmological expansion rate and that our universe appears to have entered a period of eternal accelerated expansion. The growing and shrinking of these horizons can be interrupted though if the expansion rate experiences a significant change driven by the stress-energy physics. Such is believed to have happened very early in our universe's history (too early to be visible on the plots), during and immediately following a period of rapid cosmological inflation. The early inflation quickly expanded nearby objects away from each other faster than the speed of light, creating a narrow event horizon and Hubble sphere if inflation had continued forever. But at the end of this early inflationary period, the expansion slowed considerably and the Hubble sphere and event horizon began to grow again, counterintuitively bringing regions that had fallen out of contact with each other back into observable contact again. This is an example of how a horizon can be reversed using the physics of stress-energy, though keep in mind the specialness of the unbounded cosmology.

Implications for Warp Drives:

We have seen that horizons come in several different varieties and have a range of behaviors. We have seen that some horizons can be turned on and off and even escaped, while others are much more strict. So what does this mean for warp drives? 

While we cannot exactly match the horizons we see in the superluminal warp drives of my own paper Breaking the Warp Barrier (also at https://arxiv.org/pdf/2006.07125.pdf) or others in the literature, we do see that of the horizons listed above, warp drive horizons most closely resemble the ergosphere of a rotating BH. This should not be too surprising as the warp drive geometries are formed from the same shift vector components of the metric that describe the frame dragging effect of the BH, and the ergosphere is also formed where frame dragging grows to the point where comoving observers exceed the speed of light.

That the horizon of one of these warp drives can support communication by being entered and escaped like an ergosphere, and can be turned on and off remains to be seen. Detailed calculations will tell, but I take these as promising hints that a future more carefully designed warp drive solution will be controllable at superluminal speeds. 

Happy belated New Year,

Erik

Update 1/21/22:

It was brought to my attention that some readers are having trouble finding a statement of the horizon problem. I have expanded on my non-technical statement of the horizon problem and typeset that text in boldface to make it more visible.