Einstein's contribution was fundamental and profound, a revolution in the way we think about the physics of motion, and in particular the motion of bodies in a gravitational field. This revolution began with the publication of Einstein's special theory of relativity in 1905, when Newtonian physics was well established, and most scientists believed their understanding of the physical laws of nature was complete. After all, newtonian physics had reigned supreme for something like 220 years! This blow to the scientific establishment was all the harder to take, as Einstein's interest in physics was a hobby at the time; his job was that of a patent clerk in an office in Berne. However, his new physics took the scientific community by storm.

A cornerstone of Einstein's work was an appreciation that the arena in which all physical events take place is a four-dimensional world called spacetime. In other words, to describe the location of a physical event—for example, the impact of an apple on the ground—we need four numbers, three defining its position in space, and another giving the time. In Newton's physics the three-dimensional spatial world and time were considered to be independent and absolute. However, in Einstein's theory, space and time are inseparably interwoven, and the place and time defining an event are not absolute but depend on the state of motion of the observer. This rather strange notion led to the uncomfortable idea that Newton's physics was incorrect; however, the differences between Newton's and Einstein's descriptions of the world manifested themselves only when things moved at very high speeds, that is, speeds near the speed of light of 300,000 km per second (186,000 miles per second).

Einstein's revolution was not complete, however, as in 1916 he published his theory of gravitation—the general theory of relativity. The journey from the special theory to the general was not an easy one, and Einstein struggled with the physics and, in particular, the mathematics required to formulate his gravitational theory. Indeed, the mathematics required to describe his theory of gravity were so complex that it was claimed that few people in the world actually understood it when first published. Fortunately, the principles of the theory can be explained in relatively simple terms.

Einstein's description of the way planets moved around the Sun is completely different from Newton's view. In Einstein's theory, the four-dimension world of space-time is not just a background reference system against which the locations and timings of physical events are recorded, but rather it becomes a dynamic entity, playing a central role in the way things move in a gravity field. The underlying principle of Einstein's general theory is that massive objects, like the Sun, distort the geometry of space-time. This is the famous warped space, which has become so familiar to us all, courtesy of popular science-fiction epics like Star Trek. However, although we have heard a lot about it in sci-fi stories, nevertheless an appreciation of what a curved four-dimensional space-time continuum means is very difficult to grasp, even for those equipped to understand the mathematics! Einstein's basic idea of motion in a gravity field is that objects move in such a way as to take a path that gives the shortest distance between two points. Clearly in our everyday experience, the path defining the shortest distance between two points is a straight line. But then, in our everyday experience, we do not often come across warped space!

However, there is one everyday example of determining the shortest distance between two points in a curved space—that is, the efficient global routing of aircraft. For example, what is the shortest distance between London and Sydney in the curved two-dimensional space we call Earth's surface? If we take a map and just draw a straight line between London and Sydney (the broken line in Figure 1.11), we find that this is not the shortest route. The shortest route can be found by stretching a piece of string on a globe, holding down the ends over London and Sydney. If you then plot this route on a map (the continuous line in Figure 1.11), you will find that the shortest route is curved. A quick experiment with a globe and a piece of string will help you to check this.

Returning to Einstein's gravity, as we said above, the influence of the massive Sun is to produce curvature in the fabric of space-time surrounding it. Straight lines in this space are no longer straight, but curve along the contours of the warped space produced by the Sun. The resulting orbital trajectories are effectively those found by Kepler and Newton. Given this, the reader might ask why Einstein's complex theory of gravity is needed, when Newton does a perfectly good job already. The answer is that Einstein's theory goes further, and predicts additional effects that are particularly conspicuous in very intense gravitational fields. A good example of this is the bending of light as it passes the Sun, an experimentally confirmed effect that is not predicted at all by Newton's theory. In our everyday experience, a beam of light is perhaps the best way of defining a straight line. However, in

the warped space surrounding the Sun, the path of the light is deflected (very slightly) in response to the curvature of space-time. Another curious feature of Einstein's general theory is that when space-time is curved by the presence of a massive object, not only are the spatial dimensions curved, but the time dimension is as well; we are presented with the bizarre notion that clocks run at different rates depending on how close they are to the object!

Clearly Einstein's achievements are pertinent to our story, but we should return to the question in the title of this section: What did Einstein do for us? Well, if the "us" refers to spacecraft design engineers, the honest answer is "not a lot!'' The spacecraft missions achieved in the first half century or so of the Space Age involve space vehicles that have not achieved very high speeds, compared to the speed of light. Similarly our activities have effectively been confined to the region of space near the Sun, where very intense gravitational fields are not encountered. As a consequence, the more exotic effects of Einstein's theory do not manifest themselves, and we are left with the rather surprising conclusion that modern spacecraft engineers still use 300-year-old Newtonian theory.

There is, however, one clear example where Einstein's relativity theory does make an essential contribution to the design of a spacecraft. The U.S. Department of Defense operates a space system called Navstar Global Positioning System (GPS), which is used as a navigational aid for all branches of the U.S. armed forces. However, a lot of people reading this may have used GPS for leisure purposes—hiking, sailing, or flying—or for in-car navigation. The space system comprises a constellation of 24 satellites in near-circular orbits at heights of around 20,500 km (12,700 miles). If you have an appropriate receiver on the ground, the system will provide information about your location accurate to about 10 meters in each of the three spatial dimensions. To do this, however, each satellite must carry an atomic clock, which needs to be accurate—to about one second in every 30,000 years or so! To do the necessary calculations to find your position on the ground, your receiver must also have a clock. Fortunately, this clock need not be quite so sophisticated (or expensive!) as the satellite clocks, but it should record the passage of time at the same rate as the orbiting clock, during the short period when the receiver is doing its calculations to estimate where you are. However, the receiver clock on the ground is a lot closer to the gravitational mass of Earth than the satellite clock, and therefore Einstein said the ground clock will run slower than the orbiting clock. Over the period of a day, the combined effects of Einstein's theory cause an accumulated error of around 38 microseconds (38 millionths of a second) difference between the orbiting and ground clocks. Although this sounds small, when translated into a navigational error it amounts to about

10 km. After a day your in-car navigation system might be indicating that you are in the wrong town!

When the first experimental GPS satellite was launched, some engineers were skeptical about the importance of the Einstein effects, but soon realized that time warping is a reality. To overcome this problem in the current spacecraft design, the satellite clocks are manufactured with an appropriate offset in the clock rate built in.

How we have come to understand space is a rather intriguing story, and what I have presented here is an abbreviated and personal view of something that could have a whole book devoted to it. In summary, perhaps one of the most surprising conclusions to be drawn from this discussion is that modern spacecraft engineers still predominantly use Newtonian theory to design spacecraft, and to design the orbits they travel to achieve a particular destination. We will take this notion forward in subsequent chapters, where the way spacecraft are designed is discussed in more detail.

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