The Three Dimensional Universe
© 2003 David Barwacz
http://members.triton.net/daveb
Abstract
It is commonly accepted in physics today that there are at least 4 dimensions (three spatial and one time). We can measure the position of an object using three parameters and if we want to specify the time that the object occupies that position we include a fourth (time). I will now give five arguments as to why there are only three dimensions.
Argument 1
Eliminate one dimension and see what’s left.
Simple arithmetic tells us that if we want to know how many items are left in a box after one is removed from the box, we simply subtract one from the number originally in the box. Similarly, if you have 3 items in a box and you are told that one was removed prior to your count then there must have been 4 items originally in the box.
Using the above simple reasoning, we wish to eliminate one dimension from the number we perceive and see what’s left. How does one eliminate a dimension? There is one known way to eliminate time. If an object travels at the speed of light, it’s time dilates to zero and hence the dimension is not important and can be considered eliminated.
We can then, at least mentally, eliminate time. It is also known that as an object approaches the speed of light, length contracts in its direction of motion reaching zero when the speed of light is reached. In other words, the distance along the direction of motion between the object and any other object has decreased to zero. The traveling object then sees a two dimensional universe.
By eliminating time we have a simple two dimensional universe. Adding time back gives three and hence we conclude that there are only three dimensions.
Argument 2
Measure the position of a stationary point.
To completely describe the position of a stationery point in space-time one needs only know the three spatial coordinates. The time of the measurement is irrelevant until and unless the point moves.
Argument 3
Measure the position of a moving point.
Suppose that you wanted to measure the position that an object was at and the time it was at that position. You might set up a three dimensional coordinate system and use a stop watch. You would conclude that you need the 4 separate measurements and hence that there are 4 dimensions. However, Einstein would argue and appropriately so, that the only valid means of measuring the distance is by using light. You would need the light beam to either synchronize the clocks, or you could just bounce a light beam off the object, measure the time it takes for the reflection to return and know the exact distance to the object by simply multiplying half the time by the speed of light.
Knowing that you must use a light (radar) bean to get an accurate measurement you would devise a simpler method. The obvious method is to establish a plane and a direction and then measure the angles that the reflected beam returns at. Only two angles are required. Commonly called the azimuth and elevation, these angles along with the time are adequate to determine that exact position and time. The single measurement of time gives both a time and distance parameter.
Three independent parameters are all that is necessary. Once again we conclude that there are 3 dimensions.
Argument 4
If there are only three dimensions then one would expect that using physics built on the assumption of 4 would yield erroneous results. There would be anticipated components of measurable properties that one simply could not measure. Quantum mechanical properties bear this out. Any attempts at measuring the angular momentum of an electron yield at best three components: Total angular momentum, one component of the angular momentum and the time of the measurement. No other components can be measured.
Once again we are left with only three measurable components.
Argument 5
Special relativity, as it is formulated today, assumes that an inertial observer has his own four dimensional coordinate system. Any other inertial observer (not at rest relative to him) will have a coordinate system in which at least one axis is a combination of space and time as observed by the first. In other words, it is accepted that time can be part of space for everything moving relative to us, but that it is not a part of our space. A psychologist might see this as a problem but in physics it is accepted as fact.
The better conclusion is that time is in fact a part of every observers space.
It is not possible to measure, interact with, observe or in any way describe something without time. Time is a part of every measurement we make. Space is not independent of time.
It is often said “what you see is what you get”. In the case of space-time this may be a very appropriate statement. We perceive three dimensions because that’s all there are.
In the discussion to follow, I will condense everything into 3 dimensions. I will show that the Dirac matrices and SR can be completely described in the same 3 dimensional space.