dave_a_mbs
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Hi Bob-
The "other formula" is something I blundered into while doing some speculative research on subjective space time perception at Washington State. Rather than math, I'll describe it.
What I did was to obtain, through the great kindness of the analog computing lab, an assortment of little lines of random length that moved across a video screen at a known, but random rate. Then I got people to sit down and make their best guess at the length of the line.
Now Special Relativity - if we move something faster and faster, toward the speed of light, then it seems to become shorter (with respect to us observers at rest). The general relationship, as told to me by a young lady scientist with whom I didn't quite have time to get involved (which is quite another tale), goes like this:
"The faster it goes, the shorter it gets, the more it gains inertial mass."
Now we ask how come Special Relativity happens that way. The answer is that the universe is granulated, everything, including the way we perceive things, is in terms of quantum units of some kind. Just as a tree is a tree, and does not slide off into also being a bush or a vine, all objects in our world have a nominal, discrete existence, right down to a single photon that illuminates your eyes.
This physical aspect can be studied more closely by looking at Max Planck's studies of black body radiation, where he asked what happens in a hot closed container. If all the rays of light get together and mix in the manner of smoothly continuous waves, they will eventually generate x-rays, thn cosmic rays, and finally waves of such high frequencies that each wave will carry enough energy to melt down the apparatus, or leak out through the sides. But this doesn't occur. It doesn't occur because each wave is discrete, and can be expressed as a single event, with wavelength equal to some multiple of the Planck length, about 10^-35 meters long.
It turns out that this is the shortest wave that can exist in our universe, because the mass density equivalent for its energy and size causes anything of higher frequency or shorter wvelength (the same thing) to vanish into a microscopic "black hole". Thus, there is a discrete unit of time, or "chronon" that has a real, measurable magnitude about 10^-44 second, and is the sequence interval at which the fastest transition across a material object can be made, where the object is the Planck length.
Our senses are also limited. You have two properties, in this case for the visual senses, that are defined by a visual threshold of sensitivity, and by an incremental sensitivity. The threshold is simply how much light it takes to wake up your senses to the fact tht light is actually happening. With electronics it can be as sensitive as a single photon, but our eyes require several photons, the actual number varying with the specific range of color (frequency) in which we are seeing. Yellow-green is the color of greatest sensitivity, like lime green fluorescein dye.
The incremental sensitivity of the eye (and of all our other organs) is a proportion of the stimulus level. This is a generic nervous system phenomenon independently studied by Weber and Fechner who wrote nice mathematical expressions for it. The easy explanation is for our hearing, rather than sight. To tell the difference ion loudness between two sounds, they must have a two-to-one difference in power. (That's why appliances with linear knobs seem to be very sensitive near very low power levels and insensitive at high power levels. This makes a great lab experiment for psychology classes.) In other words, to hear a difference in sond level that is just noticeable, you have to either increase the sound to twice the power level, or reduce it to half the power level. This difference is just barely noticeable, so we call it a Just Noticeable Difference or JND. (If you're curious, the threshold sensitivity for hearing is so low that it corresponds to eardrum motion on the order of the diameter of a hydrogen atom - about 10^-10 meters.)
Now let's consider our senses. All our senses have some proportion at which we can detect an incremental change in stimulus level. The JND varies with the sense, so we have one ratio for sensing heat, another for hearing, another for light and so on. This is a great advantage, since it vastly increases our dynamic range for sensation in just the same way as increasing your computer screen colors from 8 bit to 24 bit gives you 2^16 more colors. It's a logarithmic scale.
In the same way, we also have rate JNDs for the input of informtion. These JNDs are the proportional increments at which we can tell the difference between the rates of two information streams. As an example, you can hear one person talk easily, but when you try to follow two conversations, it gets difficult, and three different speakers all at once usually exceeds the information rate maximum, and we get overload and chaos. (This is one of the areas in which I do a lot of my own research. For example, I ask what is the JND for our psychic senses?)
For conversations, you can discover your own JND for conversational data input by taking two speakers, get one of them talking at normal rate, and then start the other talking at a progressively faster rate until you reach the point that you can tell that you are losing bits and pieces of the conversation. The two rates then give the ratio, hence the JND for talking.
OK, using this kind of thinking, I ran my video of lines crossing the screen at various rates, and I asked people tio tell me how long the lines were. Then I plotted the errors that they made. What I discovered was that as the lines moved faster, up to a point, errors increased as if the lines were getting shorter according to the same math function as if the lines were subject to Special Relativity. Then there came a point at which chaos set in from overload.
In the same way, since we sense things in accordance with rates, while you are sitting doing almost nothing, waiting for water to boil, for the clock's hands to move, or some similar task, the reference information rate you experience is extremely small. The JND is a proportion that is the same for all information rates. It takes only a tiny change of information flow rate to sense a change of information rate. Time is sensed according to the number of JNDs of "time sense" we experience in a given moment, so in some interval measured by the clock, we tend to experience a lot of JNDs of time which we interpret to be "a lot of time".
Now consider going to a wonderful party. When we are active, talking, dancing, having a great experience at a party, then we are experiencing time at a high rate of information flow. The JND proportion is the same, but the information flow is so much greater, that the JND proportion now involves a lot of information. Thus, it takes quite a while for us to notice that time has passed. So in the same interval by the clock we expereience few JND of time. As a result it seems that time is passing at a much faster rate.
Now consider how time feels to a person with respect to age. The JND is a fixed proportion of the age of the person. When the person is very young, then the JND is a small number of days. As the person grows older, the same JND proportion increases so it takes a larger number of days to make the same JND proportion. As a result, more time passes, according to the calendar, before one JND is sensed. Thus, us old farts sense that time is zipping by while youngsters still find time to be moving slowly, and our grandchildren who are only one or two years old find a wait of more than a few seconds to be just as intolerable as I would find a wait of weeks or months.
There is also a JND of "math anxiety" which can be reduced by verbal explanations, I hope. So there's the explanation - now you can derive the equations and whatever on your own, when you have time.
dave
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