| Topic: |
Science > Physics |
| User: |
"yurik" |
| Date: |
11 Jan 2007 08:07:15 AM |
| Object: |
Geometry is not math - it is theoretical physics |
Suppose we have a physical object: rectangular triangle with the sides
3,4,5. We want to describe it. We name the tops A,B,C and at first
prescribe arbitrary coordinates: A(a,b); B(c,d); C(e,f). We have 6
unknown coordinates which have to satisfy 3 equations of distances
between the tops. If so, then 3 of coordinates we can choose as we
like: a=b=c=0. Using Piphagor's theorem we can write the equations:
d=3; e^2+f^2=16; e^2+(f-3)^2=25. From where we find: f=0; e=4. The
result is: A(0,0); B(0,3); C(4,0). Though we have another possibility:
A(0,0); B(0,4); C(3,0).
Notes: 1. We do not need the axes of coordinates but we do need a
metric tensor because Piphagor's theorem assumes Euclidean metric. 2.
At some extent the coordinates are not important (and consequently they
have no physical meaning). We do not need to drow the coordinate axes
and triangle on paper -- we are all right with numbers only. 3. Here we
used pure math to describe a physical object -- it is typical examle of
theoretical physics. In 2-d, 3-d geometry we can do it differently but
in SR (time one of coordinates) we can not.
.
|
|
| User: "PD" |
|
| Title: Re: Geometry is not math - it is theoretical physics |
11 Jan 2007 08:45:59 AM |
|
|
On Jan 11, 8:07 am, "yurik" <y...@efn.org> wrote:
Suppose we have a physical object: rectangular triangle with the sides
3,4,5. We want to describe it. We name the tops A,B,C and at first
prescribe arbitrary coordinates: A(a,b); B(c,d); C(e,f). We have 6
unknown coordinates which have to satisfy 3 equations of distances
between the tops. If so, then 3 of coordinates we can choose as we
like: a=b=c=0. Using Piphagor's theorem we can write the equations:
d=3; e^2+f^2=16; e^2+(f-3)^2=25. From where we find: f=0; e=4. The
result is: A(0,0); B(0,3); C(4,0). Though we have another possibility:
A(0,0); B(0,4); C(3,0).
Notes: 1. We do not need the axes of coordinates but we do need a
metric tensor because Piphagor's theorem assumes Euclidean metric. 2.
At some extent the coordinates are not important (and consequently they
have no physical meaning). We do not need to drow the coordinate axes
and triangle on paper -- we are all right with numbers only. 3. Here we
used pure math to describe a physical object -- it is typical examle of
theoretical physics. In 2-d, 3-d geometry we can do it differently but
in SR (time one of coordinates) we can not.
This is a largely useless post, but I'll make one comment:
The exercise that you've sketched (rather badly) is one that is used to
demonstrate that physics can be done in a coordinate-free manner. That
is, in principle you do not need to know any coordinate positions of a
collection of cities (for example), but in fact only a chart of
physical distances between cities, to construct an accurate map of all
the cities, modulo an overall parity flip or an overall rotation.
As to your comment that this can't be done in SR -- it most certainly
can, simply replacing distances with 4D intervals.
PD
.
|
|
|
| User: "yurik" |
|
| Title: Re: Geometry is not math - it is theoretical physics |
11 Jan 2007 10:13:56 PM |
|
|
This is a largely useless post, but I'll make one comment:
The exercise that you've sketched (rather badly) is one that is used to
demonstrate that physics can be done in a coordinate-free manner. That
is, in principle you do not need to know any coordinate positions of a
collection of cities (for example), but in fact only a chart of
physical distances between cities, to construct an accurate map of all
the cities, modulo an overall parity flip or an overall rotation.
As to your comment that this can't be done in SR -- it most certainly
can, simply replacing distances with 4D intervals.
PD
Still, I thought, that it is important to know that time (as a
coordinate) has no physical meaning: only 4D intervals (which is proper
time) have a physical meaning. And so the speed of light
(x-coordinate/t-coordinate) also has no physical meaning. The meaning
of speed of light is zero 4D interval.
.
|
|
|
| User: "PD" |
|
| Title: Re: Geometry is not math - it is theoretical physics |
12 Jan 2007 08:03:28 AM |
|
|
On Jan 11, 10:13 pm, "yurik" <y...@efn.org> wrote:
This is a largely useless post, but I'll make one comment:
The exercise that you've sketched (rather badly) is one that is used to
demonstrate that physics can be done in a coordinate-free manner. That
is, in principle you do not need to know any coordinate positions of a
collection of cities (for example), but in fact only a chart of
physical distances between cities, to construct an accurate map of all
the cities, modulo an overall parity flip or an overall rotation.
As to your comment that this can't be done in SR -- it most certainly
can, simply replacing distances with 4D intervals.
PD
Still, I thought, that it is important to know that time (as a
coordinate) has no physical meaning: only 4D intervals (which is proper
time) have a physical meaning. And so the speed of light
(x-coordinate/t-coordinate) also has no physical meaning. The meaning
of speed of light is zero 4D interval.
Well, in a sense, no individual coordinate has physical meaning.
However, note that certain *combinations* of coordinates do have
meaning, such as the invariant 4D interval. Likewiise the ratio of
space to time coordinates does have a physical meaning, since that
defines the light cone, which is the boundary for physical causality.
PD
.
|
|
|
|
| User: "Eric Gisse" |
|
| Title: Re: Geometry is not math - it is theoretical physics |
11 Jan 2007 11:58:51 PM |
|
|
yurik wrote:
This is a largely useless post, but I'll make one comment:
The exercise that you've sketched (rather badly) is one that is used to
demonstrate that physics can be done in a coordinate-free manner. That
is, in principle you do not need to know any coordinate positions of a
collection of cities (for example), but in fact only a chart of
physical distances between cities, to construct an accurate map of all
the cities, modulo an overall parity flip or an overall rotation.
As to your comment that this can't be done in SR -- it most certainly
can, simply replacing distances with 4D intervals.
PD
Still, I thought, that it is important to know that time (as a
coordinate) has no physical meaning: only 4D intervals (which is proper
time) have a physical meaning. And so the speed of light
(x-coordinate/t-coordinate) also has no physical meaning. The meaning
of speed of light is zero 4D interval.
Time has no physical meaning?
Coordinate time is the rate you see other clocks tick at. Are you
saying there is no physical meaning in observing another clock?
.
|
|
|
|
|
|
| User: "jem" |
|
| Title: Re: Geometry is not math - it is theoretical physics |
11 Jan 2007 08:28:09 AM |
|
|
yurik wrote:
Suppose we have a physical object: rectangular triangle with the sides
3,4,5. We want to describe it. We name the tops A,B,C and at first
prescribe arbitrary coordinates: A(a,b); B(c,d); C(e,f). We have 6
unknown coordinates which have to satisfy 3 equations of distances
between the tops. If so, then 3 of coordinates we can choose as we
like: a=b=c=0. Using Piphagor's theorem we can write the equations:
d=3; e^2+f^2=16; e^2+(f-3)^2=25. From where we find: f=0; e=4. The
result is: A(0,0); B(0,3); C(4,0). Though we have another possibility:
A(0,0); B(0,4); C(3,0).
Notes: 1. We do not need the axes of coordinates but we do need a
metric tensor because Piphagor's theorem assumes Euclidean metric. 2.
At some extent the coordinates are not important (and consequently they
have no physical meaning). We do not need to drow the coordinate axes
and triangle on paper -- we are all right with numbers only. 3. Here we
used pure math to describe a physical object -- it is typical examle of
theoretical physics. In 2-d, 3-d geometry we can do it differently but
in SR (time one of coordinates) we can not.
Here's a puzzle for you: remove 2 consecutive letters from your posting
name to get a word the describes this post.
.
|
|
|
|
|
Related Articles |
Re: "Pure math", physics, and FLT
Re: "Pure math", physics, and FLT
Re: Newsgroup survey: Relevance, prime counting, math society
Re: Newsgroup survey: Relevance, prime counting, math society
Explaining math definition problem
Math dependency logic
Math dependency logic REVISED
physics without math, a reality?
| CrackCET to Improve your VISUAL, LOGICAL REASONING, VOCAB & MATH SKILLS
Re: The Beauty of Math
Re: The Beauty of Math
Need help interpreting a physics for a math problem
Physics, math, and problem solving
math for EM study
US sci & math score lagging
|
|
|
