General Reference
General Reference is a category of work that provides the widest possible frame of reference for participants in The Nordan Symposia.
A frame of reference is a particular perspective from which the universe is observed. Specifically, in physics, it refers to a provided set of axes from which an observer can measure the position and motion of all points in a system, as well as the orientation of objects in it.
There are two types of reference frames: inertial and non-inertial. An inertial frame of reference is defined as one in which Newton's first law holds true. That is, one in which a free particle travels in a straight line (or more generally a geodesic) at constant speed. In three dimensional Euclidean space, using Cartesian co-ordinates, this means that
<math>\cfrac{d^2x}{dt^2}=0 \qquad \cfrac{d^2y}{dt^2}=0 \qquad \cfrac{d^2z}{dt^2}=0</math>
A non-inertial frame of reference, therefore, is one in which a free particle does not travel in a straight line at constant speed. For example a co-ordinate system centred at a point on the earth's surface. This frame of reference rotates around the centre of the earth which produces a fictitious force known as the coriolis force.
Overview
Frames of reference are extremely important in the realm of physics for describing all types of phenomena. Choosing an appropriate reference frame and co-ordinate system may simplify the solution to a problem enormously. Let us consider for a second a situation which is relatively common in everyday life. Two cars are travelling along a road, both moving at a constant velocity. At exactly 2pm, they are separated by 200 metres. The car in front is travelling at 22 metres per second and the car behind is travelling at 30 metres per second. If we want to find out how long it will take the second car to catch up with the first, there are three obvious "frames of reference" that we could choose.
Firstly, we could observe the two cars from the side of the road. We define our "frame of reference" as follows. We stand on the side of the road right next to the second car at 2pm. We start a stop-clock at the exact moment that the second car passes us. Since neither of the cars are accelerating, we can determine their positions by the following formulae where <math>x_1(t)</math> is the position of car one after time "t" and <math>x_2(t)</math> is the position of car two after time "t".
- <math>x_1(t)=30t \quad x_2(t)=22t + 200</math>
We want to find the time at which <math>x_1=x_2</math>. Therefore we set <math>x_1=x_2</math> and solve for <math>t</math>. i.e.
- <math>30t=22t+200 \quad</math>
- <math>8t = 200 \quad</math>
- <math>t = 25 \quad seconds</math>
Alternatively, we could choose a frame of reference situated in the first car. In this case, the first car is stationary and the second car is approaching from behind at a speed of 8 metres per second. In order to catch up to the first car, it will take a time of <math>200 \div 8</math> seconds. i.e. 25 seconds as before. Note how much easier the problem becomes by choosing a suitable frame of reference. It would have been possible to choose a rotating, accelerating frame of reference, moving in a complicated manner, but this would have served to complicate the problem unnecessarily. It is also necessary to note that one is able to convert measurements made in one co-ordinate system to another. For example, suppose that your watch is running five minutes fast compared to the local standard time. If you know that this is the case, when somebody asks you what time it is, you are able to deduct five minutes from the time displayed on your watch in order to obtain the correct time. The measurements that an observer makes about a system depend therefore on the observer's frame of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three).
It is important to note that there were a number of assumptions made about the various inertial frames of reference. Newton for instance believed in a concept known as universal time. This is best explained by an example. Suppose that you own two clocks, which both tick at exactly the same rate. You synchronise them so that they both display the exact same time. The two clocks are now separated and one clock is on a fast moving train, travelling at constant velocity towards the other. According to Newton, these two clocks will still tick at the same rate and will both show the same time. Newton says that the rate of time as measured in one frame of reference should be the same as the rate of time in another. That is, there exists a "universal" time and all other times in all other frames of reference will run at the same rate as this universal time irrespective of their position and velocity. This concept was later disproven by Einstein in his special theory of relativity (1905) where he developed transformations between inertial frames of reference based of their relative displacement and relative velocity (Lorentz transformations).
It is also important to note that the definition of inertial reference frame (defined as one in which a free particle travels in a straight line at constant speed) does not include a requirement that the inertial reference frame must exist in three dimensional Euclidean space. This was another of Newton's assumptions which would later be disproven. As an example of why this is important, let us consider the Non-Euclidean geometry of a sphere. In this geometry, two free particles may begin at the same point on the sphere, travelling with the same constant velocity in different directions. After a length of time, the two particles will collide at the opposite side of the sphere. Both free particles were travelling with a constant velocity and no forces were acting. No acceleration occurred and so Newton's first law held true. This means that the particles were in inertial frames of reference. Since no forces were acting, it was the geometry of the situation which caused the two particles to meet each other again. In a similar way, it is now believed that we exist in a four dimensional geometry known as spacetime. It is believed that the curvature of this 4D space is responsible for the way in which two bodies with mass will meet together even if no forces are acting. In Newtonian mechanics, this is explained by a force known as gravity.
