Can you see that Rindler cannot receive signals from regions 2 and 3? This is because there is no way for anything travelling less than or equal to the speed of light to get from those regions into region 1. We say that regions 2 and 3 are beyond Rindler’s horizon. In fact, region 3 is particularly isolated because it is also impossible for anyone in region 3 to receive signals from Rindler. These two regions are completely causally disconnected. Region 4 is different; Rindler could receive signals from this region, but he couldn’t send signals into it. Rindler’s situation is very different to that of Black and Grey, for whom the entirety of spacetime is causally accessible. Rindler lives in a smaller Universe than Black and Grey. By virtue of his acceleration, he has cut himself off from some regions of spacetime. The 45-degree boundary lines to his region are generically referred to as ‘horizons’ because information cannot flow both ways across them.
Previously we encountered horizons in the context of gravity and black holes. Now we see that they also appear for accelerating observers. Is there a conceptual connection between acceleration and gravity? Indeed there is, and when Einstein first realised it, he called it the happiest thought of his life.
The happiest thought
We’ve all seen pictures of astronauts aboard the International Space Station. They float. If an astronaut lets go of a screwdriver, it floats next to them. Even globules of water float around undisturbed as mesmerising, gently oscillating bubbles of liquid. Why? The astronauts, the screwdriver, the water and indeed the International Space Station itself have not escaped Earth’s gravitational pull. They are only 400 kilometres or so above the surface: just forty times the altitude of a commercial aircraft. If you were to jump out of an aircraft, you would be unwise to assume you’ve escaped gravity and not deploy your parachute. The space station is falling towards the Earth in just the same way as you would if you jumped out of an aircraft, but it’s travelling fast enough relative to the surface of the Earth – around 8 kilometres per second – to continually miss the ground. It can continue to orbit in this way with very little intervention from rockets because there is very little air resistance at an altitude of 400 kilometres. We say the space station is in freefall around the Earth; forever falling towards the ground but never reaching it. The crucial point is that freefall is locally indistinguishable from floating freely in deep space, far away from any stars or planets; that is to say, if the astronauts had no windows and could not look outside to see the Earth below, they would be unable to do any experiment or make any observation to inform them that they are in the gravitational field of a planet. This is the reason every object floats undisturbed in the space station; there is no force to disturb them, and this is the idea that Einstein famously described as the ‘glücklichste Gedanke meines Lebens’, the happiest thought of my life. It immediately suggests that there is something interesting about the force of gravity, because gravity can be removed by falling. Likewise, its effects can be simulated by accelerating. Acceleration is locally indistinguishable from gravity, and vice versa.¶¶¶ That very important idea is known as the Equivalence Principle.
Figure 3.9. Rindler’s spacecraft.
Imagine that Rindler accelerates at 1g.**** Inside his spacecraft, Rindler’s experience would be precisely the same as if he were sitting comfortably in an armchair or wandering around his cabin on the surface of the Earth. If he had no windows, there is no experiment or observation he could perform to tell him otherwise. If he reduced the power of his rocket and dropped his acceleration down to around 0.3g, he might imagine he was sitting on the surface of Mars. No other force in Nature behaves like this. It’s not possible to remove the force between electrically charged objects by accelerating or moving around. And yet this is possible for gravity. This is the clue that led Einstein to formulate his theory of gravity purely in terms of the geometry of spacetime. Gravity as geometry. Let’s explore that remarkable idea.
BOX 3.2. Extending the Penrose diagram to two space dimensions
We have been working in a world of one space dimension where our observers can only move along a line. Much of relativity theory can be understood without needing to invoke the other two space dimensions we move around in, just as we can appreciate much of Newton’s mechanics by considering things moving along a straight line. But we should talk a little about those other two space dimensions. The left-hand picture of Figure 3.10 shows the Penrose diagram of flat spacetime in 2+1 dimensions (this is the standard notation for 2 space dimensions and 1 time dimension). It looks like two cones, glued base-to-base. At any given moment in time, we can draw a ‘now’ surface, rather than the ‘now’ lines we’ve been thinking about in our 1+1 dimensional Penrose diagrams throughout this chapter. The ‘now’ surface at the junction of the two cones (Time Zero) is a flat disk, and as we move forwards in time the Now surfaces become distorted into domes just as our lines were distorted into curves.
Figure 3.10. Extending the Penrose diagram to two space dimensions. The diagram on the left is obtained by rotating the one on the right about the vertical dashed line.
The right-hand diagram in Figure 3.10 is more closely related to the 1+1 dimensional diagrams we have been drawing throughout this chapter. Our diamonds are obtained by reflecting the triangle about the vertical dashed line. The complete 2+1 diagram is obtained by sweeping the triangle around the vertical dashed line. The left-hand diagram is a complete representation of every spacetime point in 2+1 dimensions. The right-hand diagram loses some of the information because entire circles are drawn as single dots.
It’s only in the 2+1 diagram that light cones actually look like cones. In the right-hand diagram the light cones at A and B look like ‘crosses’. Note also that the dome on the left-hand diagram appears as a line on the right-hand diagram, and that the circle on the ‘now’ sheet and the circle on the dome are actually of the same radius. The dome-one just looks smaller because we are squashing space down as time advances to make it fit into the diagram.
We live out our lives in 3+1 dimensions. We can’t draw that of course, but we can imagine. The points in the right-hand diagram would now correspond to entire spheres rather than circles.
One final word: we have taken the opportunity to introduce the notation for the five different types of infinity we mentioned throughout the chapter. The vertices of the two cones labelled i+ and i– are future and past timelike infinity – the ultimate origin and destination of anything travelling less than the speed of light. The cone’s surfaces labelled ℑ+ and ℑ– are future and past lightlike infinity, accessible only to light beams or anything else that can travel at the speed of light. The circle at the junction of the two cones marked i0 marks out spacelike infinity – the infinitely distant regions of space at any moment in time.
* De Kremer renamed himself Gerardus Mercator Rupelmundanus, Mercator being a Latin translation of Kremer, which means merchant.