Everyone knows that if you’re in a vehicle and it moves forward quickly then you’re thrown into your seat. Conversely, if it stops quickly then you’re thrown out of your seat. So familiar are these effects that we barely given them a second thought.
Anyone who did bit of science at school would know that we call the effect ‘inertia’. Newton’s Laws of Motion sum it up by stating that something stationary, or something moving at a constant speed in a straight line, will continue to do so unless acted upon by a force. In other words, it’s a natural tendency of all objects to stay at a constant speed in a straight line, or at rest. Einstein’s Theory of Special Relativity treats these two conditions as indistinguishable so inertia is really just a tendency of objects to stay at a constant speed in a straight line. Let’s use the proper term for this: ‘inertial motion’, or the motion of an ‘inertial frame of reference’.
When the vehicle accelerates, a force (from the engine) is trying to make you move more quickly. When it stops, a different force (from the brakes) is trying to make you move less quickly.
OK, simple enough, right? However, very few text books explain that we really don’t know why that is. Why do all objects have this tendency? Newton may have stated the principle but he couldn’t explain ‘why’ – that’s just the ways things seemed to be.
This same effect can be used to explain why planets travel in orbits. If there were no gravity then they would fly off at a tangent (i.e. at a constant speed in a straight line). Gravity is the force that is constantly changing a planet’s direction and making it travel around the main source of the gravity, although we’ll see in a moment that Einstein’s Theory of General Relativity has a different perspective on the force of gravity.
Let’s play some mind games. Imagine you’re in outer space in a ship that is travelling with inertial motion. There’s initially no acceleration or deceleration, and there’s no gravity nearby. What happens if the ship then goes faster or slows down? You would feel exactly the same effect as you did in the Earth-bound vehicle. Hence, it’s not related to gravity. It was suggested by Ernst Mach (Mach’s principle) that inertia is something to do with motion relative to the “fixed stars”. In other words, an effect caused by the presence of other objects around us in the universe. So what if there were none? Even though we couldn’t see anything anywhere, would we still feel inertia?
A variation of this scenario is if we imagine the ship rotating – one of the possible ways of creating an artificial gravity during space travel. We would be thrown to the walls of the ship by centrifugal force, although in reality it is inertia that wants us to fly off the ship but the walls are continually pulling us back. This is all demonstrable now, and we’d see that we were rotating because we would see the “fixed stars” passing by the windows of the ship. Ah, but what would happen if there were no visible stars? There would be nothing relative to which we could say that we were rotating so would we still feel those inertial effects? If so then what invisible concept would it be relative to.
The section on Special Relativity explains that all inertial motion is relative. We cannot say what speed we’re travelling at except in relation to something else. However, acceleration and deceleration are absolute because they can be detected without an external reference point. In Newtonian physics, an accelerated frame of reference can be treated as a simpler inertial one if you introduce fictitious forces, also known as ‘inertial forces’. Examples of these include centrifugal force and Coriolis force. For instance, consider someone far out in space, again, looking out of the window of their ship at a fixed object nearby. Initially, both have the same inertial motion so the external object is not seen to move. When the ship then accelerates, the observer inside feels a force pulling him to the rear of the ship, and also sees the external object outside being pulled in the same direction.
Note that the apparent acceleration of the external object due to this fictitious force would be independent of its mass. In his Theory of General Relativity, Einstein suggested that gravity could therefore be treated as a fictitious force (‘equivalence principle’) since the acceleration it induces is the same for all masses - Remember that Galileo showed all objects fall at the same rate if you neglect air resistance. In effect, the force of gravity is proportional to the mass of the object being affected and that results in a constant acceleration.
Rotating frames of reference are rather more subtle. Once a rigid body is rotating, it will continue to rotate without the need for external forces. The fictitious centrifugal force is added to account for the fact that objects appears to be thrown outwards, although that is just their natural tendency unless a real force is restraining them. Rotating bodies resist attempts to change their speed or direction, as demonstrated by gyroscopes, and so we attribute them with their own concept of inertia (‘rotational inertia’), measured by something called the ‘moment of inertia’. We also attribute them with an ‘angular momentum’, analogous to linear momentum, that is similarly conserved. When a spinning ice-skater extends their arms they slow down because the angular momentum is the product of the product of the moment of inertia and the angular velocity – when the first increases, the second has to decrease in order to keep the angular momentum the same.
So what is the connection between linear acceleration and rotational motion? Why do they both feel the effects of inertia? Well, when examined from the point of view of a four-dimensional space-time continuum (see Special Relativity) they both involve a twisting of an inertial frame – in an (x,t) plane for linear acceleration and in an (x,y) plane for rotation. The mass of an object is a measure of its inertia and the moment of inertia is a measure of rotational inertia. Rather than thinking of inertia as a resistance to change, though, modern physics views it as an avoidance of a ‘geodesic deviation’. We described inertial motion, above, as a constant speed in a straight line. In an ordinary “flat” space (also known as Euclidean), a straight line is simply the shortest path between two points. This concept is generalised as a ‘geodesic’ so that it is applicable to four-dimensional and curved (non-flat) continuums. For instance, for a ship travelling on the ocean, the shortest path between two points cannot possibly be a straight line but it is still a geodesic.
Einstein’s Theory of General Relativity effectively explains free-fall in a gravitational field as motion along a geodesic in a curved space-time. In other words, that’s the natural motion and gravity is therefore a fictitious force. Hence, free-fall in a gravitational field is the equivalent of an inertial frame of reference in an Euclidean space-time.
This often-used diagram shows a two-dimensional area of space around a large mass as a sheet. The change in the geometry of the sheet is represented by it being stretched in a third dimension. That extra dimension doesn’t necessarily exist, and it isn’t required by the relevant mathematics, but it helps to convey the overall effect.