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Massvolume vs. Spacetime

Zhigang Suo's picture

Apples and oranges. Each element in a set is a pile containing some number of apples and some number of oranges.  Adding two piles means putting them together, resulting in a pile in the set. Multiplying a pile and a real number r means finding in the set a pile r times the amount.  We model each pile as a vector, and the set as a two-dimensional vector space over the field of real numbers.

A vector represents different objects as a single entity. A pile containing some number of apples and some number of oranges is a vector. The addition of two vectors does not require us to add apples and oranges.  Rather, in adding two piles, we add apples to apples, and oranges to oranges.  The addition of vectors generalizes the addition of numbers:  adding two vectors corresponds to adding two lists of numbers in parallel.

Mass and volume.  We can also list different physical quantities together as a single object.  Consider a set, each element of which is a piece of some mass and some volume.  Adding two pieces means putting them together, resulting in a piece in the set. Multiplying a piece and a real number r means finding in the set a piece r times the amount.  This set is a two-dimensional vector space over the field of real numbers.  We do not have any familiar name for this vector space, and will call it massvolume. 

Spacetime.  When we list apples and oranges together, or volume and mass together, the results do not surprise us.  But when Einstein and Minkowski listed directed segments in space and directed intervals of time together, the result was shocking.  Minkowski said, “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”  What makes spacetime, but not massvolume, so shocking and so enduring?

I'm curious how you think about it. 

Comments

Hi Zhigang,

This may not what you are getting at but for you specific example of massvolume, at least there is a well known intrinsic physical quantity connecting mass and volume (the density).  Even before people knew about atoms and densities, it was probably obvious to most people that there is some sort of connection - if I have a big (large volume) rock, it is heavier (weighs more on earth, i.e. has more mass) than a small one.  Such a connection between space and time is less intuitive.  Does your argument (or question) apply to any two quantities?  For instance, would it be shocking to you to define temperaturelength?  Of course, for an ideal gas PV=nRT provides a relationship between temperature and length through other physical constants.  I guess my point is that when people hear the term spacetime, they tend to think of the connection between space and time, which was not intuitive to most people, at least before Einstein.

Zhigang Suo's picture

Indeed, mass divided by volume is density.  But derected segment devided by the interval of time is velocity.  What makes the two examples different?

Hi Zhigang,

Of course you are correct that you can take any two base units and make another unit.  Perhaps you are mainly posing a question in terms of mathematical construction in which case my response is irrelevant.  I am thinking more in terms of physics.  When I hear the word spacetime (in the colloquial sense of the word), I guess I think of the combination of space and time.  By this I mean, is time independent of space (or motion)?  Or does time progress at a different rate in a different frame of reference?  Experiments have shown that the latter is true. For instance, clocks on the space shuttle orbiting earth move slightly slower than those on earth due to higher speed of the reference frame on the shuttle relative to that of the reference frame on earth.  I personally do not find this result intuitive and perhaps others do not as well; hence, the enduring legacy of the concept of spacetime.  But, like I said, maybe you are asking something more specific to mathematical construction, in which case I do not find the result "shocking."

Zhigang Suo's picture

Dear Matt:  Thank you.  You did exactly what I would like to know:  I was curious about how people think about relatinging real-world phenomena to the vector space in linear algibra.  I was not looking for answers about linear algebra or phenomena.  

In modeling a real-world phenomenon with mathematics, we deal with two things:  the phenomenon and the model.

In the case of spacetime, the model (i.e., the relevant linear algebra) can be stated in a few lines:  A 4-dimensional vector space over the field of real numbers, with an inner product of the form x^2 + y^2 + z^2 - t^2.  The model also includes the change of basis that preserves this inner product.

The phenomenon of spacetime has many aspects:  speed of light, dilation of timecontraction of length ...

I am interested in how people think about the relations between the model of spacetime, and various phyiscal aspects of the spacetime.

In particuar:  I am now asking two specific questions.  

Dear Zhigang,

Very interesting comment! I think the problem you raised is closely related to the discussions in the previous thread (http://imechanica.org/node/15857).

The following is my thought. I think that when we deal with quantities in physics or mechanics, we should first nondimensionalize them.For example, when  F=ma
is considered, we should make  m’=
m/m0,  a’=a/a0 first,  where m0=1Kg
a0=1 m/s^2.Then we have   F=ma=(m0*a0)(m'*a').  Note that here
both m' and a' are PURE NUMBERS.  Now, mathematics tell us what the result
of  m'*a' is (e.g., 3x5=15) while physical principle
(Newton's first law) tells us what  (m0*a0) is (N, for the considered
case). In fact,

 one of the essential roles played by the physical principles
is just to tell us 'unit apple' *'unit orange' is what. From this sense, it seems better only to consider the mathematical structures (vector space, dula space, etc) of the pure number part of a phyical quantity and not the quantity as a whole

Zhigang Suo's picture

Thank you Xu for the comment.  Yes, I am still trying to update my notes on tensors.  My post above was copied from the notes in progress.  

I agree with everything you have said, but I don't think they tell the difference between massvolume and spacetime.  Here is another paragraph from my notes in progress, whcih seems to relate to what you are talking about.

Gold and silver. Each element in a set is a piece containing some amount of gold and some amount of silver. Adding two pieces means putting them together, resulting in a piece in the set.  Multiplying a piece by a real number r means finding in the set a piece r times the amount. This set is a two-dimensional vector space over the field of real numbers.  Each vector is a gold-silver piece.

A basis of this vector space consists of two pieces in the set. To ensure that the vectors in the basis are linearly independent, we pick two pieces with disproportional amounts of gold and silver. Here is a basis: e1 is a piece containing 1 gold atom and no silver atom, and e2 is a piece containing no gold atom and 1 silver atom. Here is another basis: f1 is a piece containing 2 gold atoms and 3 silver atoms, and f2 is a piece containing 5 gold atoms and 7 silver atoms.

I then went on to talk about the usual business of change of basis, and the associated change of components of a vector. 

Dear Zhigang,

       For the Gold and silver (V_GS) vector space you mentioned in the post, can we define the norm of an element in V_GS?

          If the answer is yes and we deal with the norm  in the conventional way, it seems that we should give the definitions of 

       gold*gold and sliver*sliver first... Anyway, I think whether these operations are meaningful can only be answered by physcial principles. 

       If the answer is no,  then the property of V_GS is not good enough since it only has algebric structures and even cannot be euipped with

       a norm.  

    

Zhigang Suo's picture

Dear Xu:  You ask, "can we define the norm of an element in V_GS?"  This is an interesting question, and also a practical question.  One can always define a norm, but we wish to define a useful norm.  

Here I paraphrase your question into a more generic one:

Given a vector space, can we define a map that turns each vector into a scalar?

Thus, I will not confine myself to quadratic forms.   I will also consider linear forms that turn a vector into a scalar.

Gold-silver space.  For the vector space of silver and gold, we have a very useful scalar:  the cost.  Each vector is a piece that contains some amount of gold and some amount of silver.  Suppose we determine the cost of each piece in a simple way:

Cost of each piece = (Price of gold) times (amount of gold) + (Price of silver) times (amount of silver).

This linear form turns each vector (piece) into a scalar (cost).  This linear map is clearly useful.

Massvolume.  Consider the cost of shipping some goods.  Both volume and mass contribute to the cost.  The post office must have some way to turn mass and volume into a cost of shipping.  This map may or may not be a linear map. 

Now here is a dubious application of this idea.

Rank universities.  Each university has many attributes, and a map turns these attributes into a rank.  The practice is dubious in so many ways, but at a basic level it fulfills a percieved need:  map many attributes to a single number.

We also have our own thoughts about ranking people... 

It is often really hard to come up with a useful metric to turn a list of attributes (a vector) into a number (a scalar).  It is wonderful that our spacetime has a simple, useful metric:  the Lorentz metric.

Zhigang: What makes Minkowski/Einstein's space-time special is the presence of a physically meaningful metric (i.e. the spacetime manifold is a Riemannian geometry). Restricting attention to special relativity, we can more-or-less think of the manifold as a vector space - the important thing is there is (through the speed of light) a physically relevant inner-prodcut in the space, so that the length of a vector purely in the time-like subspace can be compared to the length of a vector purely in the space like subspace and things in-between.

I suppose if one could come up with a  physically meaningful metric in your mass volume space that would be just as remarkable. 

Zhigang Suo's picture

Thank you very much, Amit!  I agree with you.  When I learned linear algebra in colleage, we went through inner product too smoothly.  The inner product was positive definite, and sounded naturual.

Most daily applications of linear algrbra are of the the type like apples and oranges, mass and volume, gold and silver.  These vector spaces do not have a "natural" inner product.

Then we have the Minkowski space, which has an indefinite inner product.

Arash_Yavari's picture

Dear Amit:

Just a minor comment. To be precise, space-time is not a Riemannian manifold; it is a pseudo-Riemannian (semi-Riemannian ) manifold because space-time metric is not positive-definite, it's just non degenerate.

Regards,
Arash

Yes, of course, Arash, agreed.

Zhigang Suo's picture

Thank you Arash and Amit.  I'd love to hear your views on the following:

  1. Why can we model the spacetime with an inner product?
  2. Why does the inner product have the +++- form?

would do a better job than "apples and oranges" for illustrating the point. However, when thinking about "mass" and "volume", Measure Theory seems to offer the best framework (and not vector spaces).

Arash_Yavari's picture

Dear Zhigang:

Neither mass nor volume can be negative numbers. So, in this sense thinking of vector spaces would be questionable?

Regards,
Arash

Zhigang Suo's picture

Thank you Arash.   It can be a concern, but might be a resolvabe concern.  Here is a paragraph from my notes in progress.

A set of gold. Each element in a set is a piece of gold of some amount. Adding two pieces in the set means putting them together, resulting in a piece in the set. We have noted before that this set is not a number field, because we do not have a sensible definition of the multiplication of two pieces in the set. However, we can readily define the multiplication of a piece and a number: multiplying a piece by a real number r means finding in the set a piece r times the amount.  We model this set as a one-dimensional vector space over the field of real numbers. We do so with some caution. The definition of the vector space requires that a piece times any real number be still a piece in the set. If the number is too large, we may not have that much gold. If the number is too small, we may reach subatomic dimension, in which case the “piece” is no longer gold. Gold is made of atoms; they come in discrete lumps. This physical fact seems to force us to use integers, but integers do not form a number field. Also, the definition of the vector space will require that negative amounts gold be in the set.
In representing a physical phenomenon with a mathematical model, we may choose to ignore these inconvenient truths initially. Once we obtain a prediction, we can check what it means. For example, if the model gives a negative amount of gold, it means that we are in deficit.

There's one man, among the greatest who walked on this Earth, who, like Einstein, put a lot of thought into these questions, a true truth-seeker: Poincare. He wrote a series of philosophical articles and essays on space, time, matter, and other fundamental questions. Everyone would benefit from reading "Science and Hypothesis", "The relativity of space", etc.

Zhigang Suo's picture

Dear Stefan:  Thank you for this note on Poincare.  I should have been clear about the context of my question.  I am preparing notes on linear algebra.  The intended readers are graudate students.  I would like to know about how students and their teachers think about linear algebra and its applications.

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