Could someone please explain the geometrization process?

  • #1
While reading about Minkowski's description of the fourth dimension time axis, I saw that by taking the product of the speed of light and time, he was able to give it a "spatial" geometric character of distance (i.e. light-years, light-seconds, or whatever). I then saw that he set c (the speed of light) to 1 in order to eliminate having to label the axis ct, giving it the label of just t for time. Since the speed of light (in a vacuum) is 300,000.000 m/s, how can a scientist or mathematician just decide to set it to a value of one? I am not a physicist nor a mathematician. I am just an aging 68 year old man that is curious and likes to learn new things. Can anyone (young or old) explain this to me? Thanks in advance.
 

Answers and Replies

  • #2
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By setting the speed of light to one we don't actually mean that we change how fast light moves. We change the system of units we measure everything in. The description in SI units is completely arbitrary, we might as well just use another system. In natural units (where c is set equal to one), velocity is dimensionless and time has the same dimension as space, namely length. We chose it that way because it simplifies calculations considerably. When comparing results with measured data however, one should not forget to convert those results, so that one can make sense of calculations in comparison with experimental data.
 
  • #3
You say "convert the results". Do you mean that I have to remember that I divided c by 300,000,000 to reduce it to 1, so I have to multiply the results by 300,000,000? I understand the need for simplicity and I understand the dimensions of space, I just don't quite understand this process. Can you plug in some numbers and show me what you mean in a real life example? Thanks.
 
  • #4
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lsimpson1943 said:
Do you mean that I have to remember that I divided c by 300,000,000 to reduce it to 1, so I have to multiply the results by 300,000,000?
Correct.

Hi Isimpson1943,

I am another 1943. What you can take away from the concept of setting C equal to unity without units is that there is an error somewhere. Units do matter. The units of velocity are meters per second. The units of time are seconds. Only forced results require steps that would change either of these two. I am not explaining here, because it is unwelcome here, about forced results. What I will say is that there are choices to be made in theoretical physics that begin at its beginning, such as with making mass an indefinable property. An indefinable property is one that cannot be defined in terms of pre-existing properties. When this occurs, indefinable units must also be introduced. The units of empirical evidence are those of meters and seconds. When you see those disappear, you are correct to question what is going on?

James
There is no error in setting c equal to unity and dimensionless. There is in principle nothing which keeps us from measuring time and space in the same units. It's just that the SI system is the traditionally established norm in the world of natural sciences. The concept of length and time is empirical, not the units we measure them in. Mass is in no way more or less fundamental than for example length. Please explain your concept of "forced results", since you apparently use it as an argument.
Every system of units is fine as long everybody agrees to use the same system when comparing theoretical results/experimental data.
 
  • #6
Thanks, Polyrhytmic. I read the article in wikipedia for which you had provided a link. It cleared it up for me, I think. If I understand everything covered this far correctly, geometrization involves setting certain recognized constants in Planck units to 1 in order to simplify equations. The person doing the calculations then must reverse the simplification process when the final math is worked out. Like in this case where distance = ct was reduced to distance equals time, I must multiply the final math by a factor of c in order to come up with the correct magnitude.

It also appears to me, in the case of the Minkowski's use of time as the 4th dimension, that this process is used to allow a temporal measurement like time to be plotted in a coordinate system as if it is spatial in nature. In this manner, we are comparing apples with apples. Correct?

Sorry for asking what probably appears very elementary to the hard core mathematicians and physicists. Thanks for your patience.
 
  • #7
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You've understood it correctly, apart from the fact that the use of different systems of units is not called "geometrization". Geometrization was the historical process of explaining physics in geometric language (for example spacetime). The fourth axis in a Minkowski diagram is called ct because it gives the correct relations. One might only label the axis t, but one should not forget that the c is actually there and should be considered when talking about results.
 
  • #8
James, since you and I apparently are the same age, one thing we can definitely agree on is that it takes us a wee bit longer to catch on to complex concepts than when we were in our prime. At age 68, I find myself rereading information over and over before I begin to understand it.

You and I were 12 years old when Einstein died, and we were not taught about the Big Bang even by high school age. And if you were like me, you knew very little about relativity by the time we were graduating from high school. I went on to college and pursued a career far removed from science and math. When I retired a number of years ago, I decided to dedicate some time working with my local middle school system, in whatever capacity they needed me. It is amazing what these kids are learning in math and science at such a young age compared to you and me. It has really opened up my eyes and mind, and has made me want to learn more. Reading about super strings and m-theory with talk about 11-dimensions made me feel an unacceptable level of ignorance. I am correcting that feeling now, but it has required me to think way outside what would normally be my comfort zone. My sense of reality has definitely been changed forever.

I understand your concern about the possibility of using unbridled practices to justify science equations. If I thought that were the case, I certainly would find it unacceptable. However, in this case, I am just thinking my lack of understanding the geometrization process is the real culprit. I know there are people a lot smarter than me making up the rules of the game. Sometimes I just need a little extra help to think like they do.
 
  • #9
Polyrythmic, thanks again. I will correct my use of the word "geometrization" in the future. You have been very helpful, and I truly appreciate it.
 

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