What is a Vector that Curves? [possibly a stupid question]
PoundSign2
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Today while browsing the internet I came across some pictures of space. So naturally I followed up random Google images by going to NASA's website and looking at some really fascinating stuff. While there I saw a photo of gravitational lensing. Now I'm not going to explain what that is, I'm just going to illustrate the point and role this picture played.
Tuesday happens to also be the day I go to campus for a math course. This got me to thinking, "If vectors have direction and magnitude, such as a photon (traveling through x,y,z @ c [speed of light]) and a photon undergoes gravitational lensing. Then what is it called when the vector undergoes a continuous change in direction?"
Now my gut tells me not to be an idiot and just call it a curve, or a bend, or something simple. But my brain in all its wisdom says there's something interesting to be had here... Although I am curious if gradients play a part in this sort of thing. Is there a mathematical concept behind vectors that might curve or do something other than travel in a straight line? I'm also not sure if perhaps you define a continuous curve for a vector as something similar to a tangent line. How it's straight at any given point (the tangent line) but the overall effect is a curve? Anyways, any thoughts on this?
Tuesday happens to also be the day I go to campus for a math course. This got me to thinking, "If vectors have direction and magnitude, such as a photon (traveling through x,y,z @ c [speed of light]) and a photon undergoes gravitational lensing. Then what is it called when the vector undergoes a continuous change in direction?"
Now my gut tells me not to be an idiot and just call it a curve, or a bend, or something simple. But my brain in all its wisdom says there's something interesting to be had here... Although I am curious if gradients play a part in this sort of thing. Is there a mathematical concept behind vectors that might curve or do something other than travel in a straight line? I'm also not sure if perhaps you define a continuous curve for a vector as something similar to a tangent line. How it's straight at any given point (the tangent line) but the overall effect is a curve? Anyways, any thoughts on this?
Comments
C.W.
Newton invented the differential calculus to attack the tangent to a curve problem. http://en.wikipedia.org/wiki/Differential_calculus Basically he came up with a way to divide zero by zero and get an sensible answer, the slope (tangent) to a curve at a point on the curve.
You can extend that to 3 dimensions. Imagine the field lines of a bar magnet. I'm sure you have seen those diagrams of field lines coming from the north pole and travelling to the south pole. At any point in space that field has a magnitude (strength) and direction. It's a vector. But it continuously changes as you move through the space. It's a "vector field". http://en.wikipedia.org/wiki/Vector_field
I just have to nod my head and smile when my father-in-law refers to the horse-shoe driveway as curvilinear.
Oh wow the magnetic field seems kind of obvious now that I think about it. The vector fields look pretty interesting, I like how it describes the curve as tangent lines & vectors. Thanks Heater & ctwardell for sharing that information.
There's a nice little book on the subject. Get a used copy for about $10 or borrow it at the library.
http://www.amazon.com/Div-Grad-Curl-All-That/dp/0393093670/ref=sr_1_3?s=books&ie=UTF8&qid=1386123286&sr=1-3&keywords=div+grad+curl+and+all+that
For a velocity vector, I think the answer would be an acceleration vector, which is what a gravitational field looks like to a mass.
-Phil
-Phil
The minimum set up is:
1) Emitting a photon from point A.
2) Detecting a photon at point B.
3) Nothing but free space between and around A and B.
Now, we like to say that the photon travelled from A to B in a straight line.
Clearly we have no way to tell if it did or not. And as you know if we put something in the path or possible paths to find out then we don't have free space any more and diffraction effects come in to play, at which point the path will for sure not be straight.
Worse than that, mathematically if we want waves to get from A to B we have to add up the resulting amplitudes and phases of all the possible paths between A and B, through out all space! They are all interfering with each other to give a maximum at B. That is anything but straight line travel.
There is a nice video on youtube where Richard Feynman introduces Quantum Electro Dynamics that makes this, sort of, clear.
I'm just thinking.... What if you emit a photon at point A, let it travel 100000 light years until it passes through point M, let it travel another 1cm until it travels through point N, then let it travel another 100000 light years before you detect it at point Z. Is it safe to say the photon went in a "straight line" through points M and N?
Actually, a "straight line" in relativistic terms is referred to as a "geodesic." From Wikipedia:
-Phil
But can't you also think of a photon as a probability wave? And its direction is straight? Where it collapses into "reality" is just a function of its probabilistic distribution.
Ah, math models. I can't find the quote, but I remember somebody asking Feynman about the apparent negative arrow of time that fell out of one of his theories. He just shrugged and said something about it being merely a convenient mathematical construction. "Who the hell knows what it really means?"
@Mark_T You still would be right, though?
Erlend
ElectricAye, We did not bring QM into this but let's see:
You really can't think of waves as travelling straight anywhere. Every point on the wave front at any given instance contributes to all points on wave front the next instance. In all directions, including backwards. The sum of all those contributions determines what the wave looks like next. Or at least that is the way Richard Feynman puts it in his "Quanum Electrodynamics For Dummies" book (or whatever it's called).
Consider a photon emitted from an atom at A and arriving at atom B. There is also an atom C where it could end up in this little universe.
From a quantum mechanics point of view a wave is emitted from A and goes on to permeates the entire universe. That gives rise to a probability that the energy of the photon ends up being absorbed by atom B or C. Perhaps a 50% chance each if they are equidistant from A.
We might think of the "wave" being emitted from a point and as such travelling out in all directions. Just like the ripples caused by a small stone dropping into a still pond.
Having explored the entire space, in all directions, at the same time, and discovered all possible landing sites for the photons energy something magical happens and the photons energy is sucked in my atom B or C. The so called "collapsing of the wave function"
Nothing actually collapses anywhere except the mind of the physicist thinking about it:)
Is there anything "straight" about the above description? The wave part? The particle trajectory part?
None that I can see.
Well you could say that, but really we only know how to calculate the probability, not the mechanism
by which it works, since that is beyond experiment I believe - my statement above "from the point of view
of a photon" is obviously suspect since its hard to see a mechanism by which a massless particle can have
a point of view at all. I don't know enough to comment on relativistic quantum mechanics.
Our intuition is often wrong with quantum machanics because of the way _all_ possibilities have to be summed
and multiplied as complex numbers, and our expectation that because we see a particle arrive at a detector
it actually took a specific route to get there, whereas all the theory says is the probability of the outcome.
Newtons Laws of motion, well that that describes what happens in terms of things called mass and inertia. It says nothing about what they are. Why should an object continue moving if nothing is pushing it?
Gravitational Laws, pah, says nothing about what gives rise to gravity itself.
QM, more of the same, we have some maths, we can measure the probabilities, it seems to fit. Why? Who knows? Quite so.
Actually I was thinking again on my description of the universe with three atoms in it and a photon jumping from atom A to B or C.
Of course what actually happens is that the energy of the photon is at A or B or C with some probability, perhaps one third each if they are equidistant. After all there is no place else in this three atom universe for it to be. It cannot be said to "be" at any of those locations, it is at all of them.
Or to look at it another way, that photons worth of energy that jumped from atom A to B had no particular reason to have done so and having got there no particular reason to not to jump back to A or perhaps move on to C.
Or, that's how I see it today. Might be different tomorrow.
That reminds me of a video by SixtySymbols or Numberphile, I cannot remember (they all generally have the same professors) but they discussed how light travels through glass and the points/paths it takes. To sum it up rather quickly, the light is said to take all paths, at the same time. The Uncertainty Principle just tells us where we can view it at any given moment. Or something to that affect. =]
Since we're on the subject of space and all things math and science, one thing that's always baffled me is this concept that space-time is this linear grid. Where space/time is this fabric and the indentations are gravity. Like this...
It makes no sense in a 3D World because indentations would be everywhere, all the time. In this picture, where the massive cavity caused by the sun, or the moon? I don't think the grid example is very accurate. To me space-time is like goo, and gravity is like convection currents where more massive objects have more convection currents and a stronger 'flow'. It might be outlandish but I think my brain processes it like that easier than viewing a flat, planar grid of a 3D object.
What do you think?
SixtySymbols and Numberphile are brilliant.
That business about light permeating all paths in order to arrive at a destination is what I mean when I say it never travels in straight lines.
I can't really comment on this curved space time business. I like my space to be straight and on a nice square grid
For sure the stuff in the space seems more like goo than any particles and such.
They are great! Vsauce is also excellent, and SciShow as well. Those make up about 80% of my time on YouTube. haha.
Being that we cannot prove what causes gravity, just that gravity exist, I'd say my goo & currents hypothesis is about as solid as some other proposed ideas and concepts about gravity. (Notice I did not say theory! ha!)