DrRocket

Cool, thanks for the confirmation.

 

But how is that rate created? The field strength? It seems intuitive now that field strength would determine how many Joules of work would be required to move a single unit of charge toward the source or the high energy potential. Seems it would be that way for all field forces.

All force fields define the work expended to move an object between two points along a given path.

But only fields that arise from a potential, conservative fields, are such that then points alone determine the work; i.e. the work is path-independent and is 0 for all closed paths.

Not all fields are potential fields. The electrostatic field is a potential field.

Cooper pairs are when two fermions "act as one" to form a bosonic quasi-particle. This does not have to be two electrons. But yes, Cooper pairs are essential in standard superconductivity as well as the superfluidity of Helium 3.

But it is rather difficult to have superconductivity (electrical) unless there are electrons around, so I think the reporter misinterpreted the significance of the Cooper pairs reported by the scientists.

"What if these primitives that we take for granted shouldn't be taken for granted. Do they bear further investigation ?"

 

Sure.

But in the meantime taking time and space as primitive is the best that we can do.

Any more fundamental, and useful (not a bunch of philosphical mumbo jumbo), definitions would be a great step forward.

My understanding is that lattice (discrete) theories have been attempted but have failed.

If you can build better predictive theories than GR and QFT on some new concept of time and space that would be terrific. People are trying. Results thus far are underwhelming. Ver Linde has a pre-print in which he sees "space" as emergent from information -- but I see way too much vague hand waving in his "logic", and it has been awfully quiet since he gave his talk..

My problem with rest mass for composite systems is where do you stop? You're going to have energy bound up in bonds of various kinds as well as thermal, pressure etc. You'd need to go to all the effort of figuring out how much that is (yes, I know it's usually <<1%, but it's the principle that irritates me) rather than weighing it.

Most of the time, for a composite system we don't really have rest mass and energy. Instead we have some of the energy, and some more of the energy.

To me this means that neither has any right to conceptual primacy.

On the other hand, calling both of them mass is annoying to the highest degree as well.

Maybe we need a new word for total energy when you put it in a box, and stop the box? Rest mass is often used for this, but then when you open the box and only photons fly out you suddenly have to rename it.

</bugbear since 1st year when we were asked a question about mass after being told about relativistic mass, but before anyone had mentioned rest mass>

I suspect that the problem goes way back to very early science classes where everyone was exposed to the littany differentiating between mass and weight (no arguement there) and being inculcated with the notion that mass is fundamental.

Mass is not quite so fundamental as folks were told in those early days. It just is not that easy.

What you measure on a labratory scale is relativistic mass, in the version of "invariant mass" taken in a reference frame of zero momentum.

The mass that shows up in general relativity is also relativistic mass, but momemtum flux also shows up in the stress-energy tensor, and curvature is independent of any net linear velocity. So there is not a direct and easy correspondence between just rest mass or just relativistic mass either. GR is very subtle.

In quantum field theories it is natural to consider mass as a charactristic of a particle and rest mass is the natural concept. What is really fundamental is the perspective that mass and energy are the same thing, and what is conserved is neither (rest) mass nor energy but mass/energy as a single entity.

I suspect that the rigid "mass is rest mass" stance that is common these days comes from the particle physics influence. There are a lot more particle physicists than relativists.

Perhaps you are right and we need a different word, or words. We are somewhat prisoners of our language, and "mass' is with us. It just is one of those words with several definitions. It is not a big problem as long as one is clear on the definition used in any given situation.

Misner, Thorne and Wheeler attempt to break the language straight-jacket by using words like momeneregy, mass/energy, and of course spacetime. That does sometimes help, but their terminology is not widely used.

Similar terminology issues exist in mathematics -- compact vs the French (compact + Hausdorf), Riemannian vs Pseudo-Riemannian, etc. -- but mathematicians seem more accepting of adopting local author-dependent conventions, perhaps because mathematicians are very careful about the definitions that are used in any specific theorem.

For composite systems "relativistic mass" could be a useful notion, but I don't see how it is really any different to the total energy (in a specified frame), as stated by swansont. (The same goes for mass of a space-time in GR.)

It isn't any different. The terminology does reinforce that mass and energy are the same thing, two sides of a single coin.

What about rotational motions, like the rotation of a star? Are you saying, that a (hypothetical) non-rotating star, would produce the same curvature, in spacetime, as a (realistic) rotating one ? If the SET 'looks like' velocities factor in, then how is it 'invariant' ? And, again, what about 'non-inertial' velocities, like rotations, which everybody would see the same, and which everybody would identify as different, from a static situation ?

There is a rotational effect, the Lens-Thirring effect, aka "frame dragging" associated with a massive rotating body. I don't know much about it.

What about motions being 'invariant' ? What makes 'thermodynamic motions' immune to the 'invariance' of the SET ?

????????

I've got myself turned around apparently. I thought voltage was the potential difference between oppositely charged poles - that voltage is a measure of the difference in charge carriers. I'm envisioning ionized clouds of opposite charge and I thought that the amount of voltage between the two would depend on the amount of the difference in charge between the two - that the amount of voltage would depend on the number of charge carriers.

 

But somewhere in that bit up there I'm apparently off. Because you can have high voltage with only a small amount of charge carriers, or a small voltage with a high quantity of charge carriers. I'm not getting that at all and it's making my intuition cry.

 

And I've been through the water pipe analogies, and I actually do get the "pressure" concept. But with water pipes I can see where 'pressure' comes from and I can imagine a pint of water under 200 lbs of pressure. But I cannot "see" where voltage is coming from, and I cannot imagine a trickle of current from a thousand volts of 'pressure'. (please note, I'm not talking about resistance and insulating materials that restrict the flow of charge carriers, rather I'm assuming a perfect conductor).

 

Can anyone see where I'm going conceptually wrong here?

Start with the electric field from a static arrangement of xharges.. A point charge creates a spherically symmetric electric field that drops off like 1/r^2. The field due to an arrangement of point charges

is the vector sum of the fields due to the individual chsarges.

A electrostatic field is a conservative field. Therefore it is the gradient of some scalar field, called the potential. The voltage, aka potential difference, between two points in space is the difference between the values of the potential function at those two points.

So, voltage depends on both the amount of charge ane the spatial arrangement of that charge.

 

The experiment was done in only one inertial frame (the Earth.) Well an approximately inertial frame. Anyway, the light source and the detectors of that light were at rest with respect to each other. So only one inertial frame. So not a test of Einstein's light postulate.

The Earth is not an inertial reference frame, though it is a fair approximation over short time intervals during its orbit around the sun.. The Michelson-Morley experiment was repeated at different times of the year and by various experimenters over a period of years -- different reference frames.

http://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/morley.html#c2

In my logics class today, we had our midterm. On it was a bonus that I couldn't figure out. It reminded me of something from Calc III (multivariable calculus), but I couldn't nail it down. The question went something like this:

 

Let B be a box, made from spatial dimensions L, length, W, width, and H, height. Let S be all the points around the box that are, at most, 1 unit away from the box. Express S in terms of L, W, and H.

 

How should I have solved this? I had a picture where there was a cube, with a larger, sphere-like object, because the maximums would curve around the vertices of the cube. Did I start it right?

You have the relative complement of a box of dimensions L+2, W+2, H+2 and a box of dimensions L, W, H (I assume that "around the box" implies "outside of the box) -- a solid rectangle with another solid rectangle removed from its center.

This is not so much an exercise in logic as an exerecise in converting imprecise language into precise language. One has to figure out what the speaker actually means, which can be subject to interpretation.

Edit: The comments below are correct. The outer box is actually a rectanglular solid in which the edges and corners are rounded off, with a radius of curvature of 1.

 

 

If I measure the speed of a car I use my clock, not the car's. I also use my own meter stick.

 

The slowing is relative to another frame. You never see your own clock slow down.

Right, and you don't see your meter stick shrink either.

Both you and the driver of the car will agree on the speed of the car relative to you (or of you relative to the car). The time dilation and length contraction effects compensate so that the ratio is the same in both reference frames. Were this not true the resulting asymmetry would allow identification of a preferred frame.

No, your claim was that relativistic mass shows up in the paper, and it doesn't. E=mc^2 appears in the context of being true in what Einstein defines as the rest frame.

Wrong. Read the paper. Note the reference to variable mass, noted in my earlier post, which makes no sense if one is in the rest frame of the object.

It was your claim that [math]E=mc^2[/math] applies only in the rest frame:

"Usefulness" is subjective. Relativistic mass is just a proxy for total energy, so why not use total energy?

 

[math]E^2 = m_0^2c^4 +p^2c^2[/math] is the equation you start with. [math]E=mc^2[/math] is only valid at rest; if you use it in general, you're making up a new equation, and then have to worry about what you mean by mass.

It is quite clear that this is not true, since as you have asserted [math] E^2=m_0^2c^4 + p^2c^2[/math] is valid in any inertial frame and as shown previously and below it is eqiuivalent to [math]E=mc^2[/math]

You do, of course have to worry about what you mean by mass, and [math]m[/math] is "relativistic mass".

But you have to worry about what you mean by mass in any case. If you cling to "mass = rest mass" in all cases then what you measure on a laboratory scale is not mass. What you measure on a laboratory scale is in fact the relativistic mass in a reference frame in which the net momentum of the system of molecules is 0, what is called "invariant mass". Invariant mass is not the sum of the rest masses of the individual molecules. So, using a dogmatic "mass = rest mass" approach results in a macroscopic definition of mass that is not additive -- the mass of a system, as measured in the conventional manner, is not the sum of the masses of its constituent parts. You can adopt that stance, but it is, kindly, unconventional, and contrary to normal useage at the macroscopic scale. It requires some new definition of mass for the old reliable [math] F=ma[/math] and [math]F=\dfrac {E}{c^2}a[/math] just doesn't satisfy. Replacing the more rigorous [math]F=\dfrac {dmv}{dt}[/math] with [math]F= \dfrac {d (\frac{E}{c^2})v}{dt}[/math] is equally unappealing.

I didn't say rest frame of the photon. I said the rest frame of the system, i.e. what Einstein defined in the paper, the rest frame of the massive object. The energy of the photon is not frame invariant; the frame in which the photon energy is L is the rest frame of the massive object.

Einstein assumed the system to be at rest. When he concludes that the emission of energy L reduces mass by L/c^2, one must acknowledge that L — by his terminology — is the rest frame energy of the photon. The equation is not valid in any other frame.

I misinterpreted what you meant by "rest frame energy of the photon. But the statement that "The equation is not valid in any other frame" is incorrect, as shown below.

Nothing dogmatic about this. It's a matter of what you can derive.

Yes, it is a matter of what you can derive.

And [math]E=mc^2[/math] is derivable from and equivalent to [math] E^2 = m_0^2c^4 + p^2c^2[/math] and I showed you the derivation. I'll repeat it here.

[math]E = \sqrt{m_0^2 c^4 + p^2c^2}[/math] [math]= \sqrt{m_0^2 c^4 + \gamma ^2 m_0^2 v^2 c^2}[/math] [math] = \sqrt { m_0^2c^4(1 + \gamma^2 \frac {v^2}{c^2}})[/math] [math]= \sqrt{\gamma^2 m_0^2 c^4}[/math] [math]= mc^2[/math] so long as [math]m_0 \ne 0[/math]

So, "mass=rest mass" in all situations is in fact dogmatic. Better to fit your definition to the problem at hand, being clear of the definition used. It has the advantage that you don't have to give up [math] F=\dfrac {dp}{dt}[/math] which is rather useful for macroscopic dynamics problems; you just have to accept [math]m[/math] as relativistic mass.

Lev Okun has written a few papers documenting that this is not the case. Here's one http://arxiv.org/abs/0808.0437

I don't care about Lev Okun's view of history. The mathematics is right here. There is nothing radical in what I am telling you. This is in any number of texts on relativity.

If you accept [math]E^2 = m_0^2 c^4 + p^2c^2[/math] you must also logically accept [math]E=mc^2[/math] for they are the same equation (for [math]m_0 \ne 0 [/math]).

The bottom line is that "mass" has several different meanings, all of which are legitimate in mainstream physics.

Einstein assumed the system to be at rest. When he concludes that the emission of energy L reduces mass by L/c^2, one must acknowledge that L — by his terminology — is the rest frame energy of the photon. The equation is not valid in any other frame.

That is simply wrong.

[math] E=mc^2[/math] is valid in any inertial frame if [math]m[/math] is relativistic mass.

From the Einstein paper : "The mass of a body is a measure of its energy-content; if the energy changes by L, the mass changes in the same sense by L/9 × 10²⁰, the energy being measured in ergs, and the mass in grammes."

The reference to variable mass makes it clear that Einstein was considering what is now called "relativistic mass".

BTWThere is no such thing as "the rest frame of the photon" in special relativity -- that would invoke singularities in Lorentz transformations that one cannot live with.

You can do this either way, there is no need to be dogmatic.

[math]E = \sqrt{m_0^2 c^4 + p^2c^2}[/math] [math]= \sqrt{m_0^2 c^4 + \gamma ^2 m_0^2 v^2 c^2}[/math] [math] = \sqrt { m_0^2c^4(1 + \gamma^2 \frac {v^2}{c^2}})[/math] [math]= \sqrt{\gamma^2 m_0^2 c^4}[/math] [math]= mc^2[/math] so long as [math]m_0 \ne 0[/math]

The notion of relativistic mass goes back to Einstein, and was taught as a matter of course for many years. The convention to regard mass as "rest mass" is relatively recent and not universal. In either case we are talking about a convention, and not a question of "right" or "wrong" Regarding mass as rest mass is very convenient in quantum field theories (where m_0 can easily be 0), and less so for macroscopic problems in special relativity. In general relativity "mass' is even more murky, and nobody knows what to do about quantum gravity.

"Usefulness" is subjective. Relativistic mass is just a proxy for total energy, so why not use total energy?

 

[math]E^2 = m_0^2c^4 +p^2c^2[/math] is the equation you start with. [math]E=mc^2[/math] is only valid at rest; if you use it in general, you're making up a new equation, and then have to worry about what you mean by mass.

Yes, you have to worry about what you mean by mass. That was the point.

No, I am not making up a new equation, Einstein made up [math]E=mc^2[/math] (expressed a bit differently in the first paper, but clearly involving variable mass).

http://www.fourmilab...tein/E_mc2/www/

If you use [math]E^2 = m_0^2c^4 +p^2c^2[/math] you have to come to grips with what you mean by momentum, and that will take you right back to "relativistic mass".

Using total energy is fine. But you need to know what it is. Since what it is is [math]\gamma m_0 c^2[/math] for a body of positive rest mass you are right back to relativistic mass.

Best to remain flexible and use the approach most convenient for the problem at hand.

I find the recent tendancy to be dogmatic and demand that mass be rest mass rather amusing, and limiting.

Of course philosophers are useless!

I said philosophy, not philosophers.

I agree that the science and maths should lead the way, but sometimes we should sit and ponder afterwards.

Make up stories to help teach it to others and make it more understandable. There's something both comforting and helpful to the intuition about having ontological objects to work with. Be they little ball-like particles, wiggling electrons, or universe-permeating fields.

Maybe even break down some concepts, find a little crack that we can wedge open with more maths and experiment.

 

I guess that in many ways, this is indistinguishable from doing more physics.

It's certainly not the thing that philosophers do (for the most part).

I used the word because I wanted to distinguish between These two things:

Breaking down or creating new assumptions and predicates. Along with finding new representations, stories and ontologies for theories which do not add any predictive power, merely make them easier to use. I call this philosophy for want of a better word. The kind of philosophy done by someone who is very familiar with physics.

and

Finding new mathematical objects, making new assumptions, combining predicates and assumptions, making predictions and doing experiments. I call this science.

 

I'm not quite sure where to put the making new assumptions part, it fits in both to a degree, although better in the latter. Perhaps this is why we shouldn't draw these sharp distinctions and accept that there is a continuum.

agreed

I don't know if light could ever travel in a recursive loop without some gravitational mass to curve space. I've heard some people say that light can generate gravity in itself, so maybe there's some situation possible in which light could curve into a loop due to its own innate gravity. Either way, the point is that I would like to know whether IF light would travel in a closed loop, would the loop have inertia/mass or could it travel at C as a loop? It seems to me that it would not be possible for such a loop to move at C because as it approached C, one side of the loop would be redshifting while the other would be blueshifting. It seems like particles with mass could resist force for a reason related to this.

I don't think the answer is settled, but something similar was hypothesized by Wheelere, the geon.

http://en.wikipedia.org/wiki/Geon_(physics)

 

I don't think philosophy is worthless regarding these questions,....

http://depts.washington.edu/ssnet/Weinberg_SSN_1_14.pdf

I am doing a jr high science experiment testing the heat retention of different types of coffee cups..styrofoam, paper, plastic,ceramic, and stainless steel. Is heat retention physics? Thermodynamics? What causes heat retention? I would like to make a really good report on this subject...does this involve intermolecular forces?? any websited to help with my research..I have done 2 trials of my experiment but want to beef up my research and report .. Thanks!!

he subject is called heat transfer. There are three mechanisms, convection, conduction and radiation. Material properties are central to conduction and radiation. The problem of keeping coffee hot in a cup involves all three mechanisms.

Heat transfer is definitely physics. However, many specialized texts are written for and by mechanical engineers.

Every analogy is flawed.

 

And I think we were mistaking 'proper motion' with 'peculiar motion' I am pretty sure it is peculiar motion that we are talking about in this thread.

 

Here's my understanding of 'peculiar motion'... We have assigned a geometry to space. The balloon analogy is always the easiest to work with, so we decorate the balloon with a drawn-on grid system, which represents the geometry of space (since the surface of the balloon is representing space itself). On our decorated balloon, we place a grain of sand in the top left corner of one quadrant (or square) of it's geometry. we place another grain in the top right corner of a quadrant 3 quadrants over from the first. We now define 'peculiar motion' as any motion relative to the geometry (or gridlines, or quadrants) of space (the surface of the balloon). Take a picture of the position of the grains of sand in their quadrants of 'space'. Measure the distance between the grains of sand.

 

Now blow the balloon up some more. Take a picture of the position of the grains of sand in their quadrants of 'space'... They haven't moved. One is in the top left corner of it's quadrant, and the other is still in the top right corner of it's quadrant, 3 quadrants over. By our definition of 'peculiar motion', they have not moved 'through space'. Now measure the distance between the 2 grains... the distance has grown. Without moving 'through space', the distance between the 2 grains has increased.

 

The law 'nothing can go faster than the speed of light' only applies to objects moving 'through space'.

 

Their is a clear difference between objects moving 'through space' and objects moving 'with the expansion of space itself'

Right

http://en.wikipedia.org/wiki/Peculiar_velocity

http://en.wikipedia.org/wiki/Proper_motion

If we want to talk about the "origin of space" or the "emergence of space" as a first principle - before we introduce anything inside it ... what is it that we are actually referring to ?

 

Sometimes the simplest questions are the hardest:

 

"What do we mean by 'space' ?"

 

In any theory there are some things that are taken as "primitive" and understood without definition. You cannot write a dictionary starting from the initial premise that no words are understood.

"Space" (or "distance") and "time" are such terms.

The best that we have are the operational definitions "distance is what rulers measure" and "time is what clocks measure". No better definition currently exists. Philosophers tend to obfuscate rather than illuminate when they address the issue.

More fundamental definitions could be very valuable. but I am not going to hold my breath in anticipation.

Also, mass is invariant — it doesn't increase with speed.[/modtip]

That depends on what one means by "mass", in special relativity.

The current fashion, apparently inspired by elementary particle physics, is to equate "mass" ([math]m[/math]) with "rest mass" ([math]m_0[/math]), wich indeed is invariant, essentially by definition.

But there is also the very useful concept of "relativistic mass", [math]m = \gamma m_0 [/math].

[math] E=mc^2[/math] is correct (admitting that the zero-rest-mass case must be handled separately) and easily remembered if one interprets [math]m[/math] as relativistic mass. If one works only with rest mass one is stuck with [math]E^2 = m_0^2c^4 +p^2c^2[/math]. This latter equation applies to the zero-rest-mass case, but also raises the question of what one means by momentum, [math]p[/math]. In the case [math]m_o \ne 0[/math], [math]p=mv[/math] where [math]m = \gamma m_0[/math] so one still has to deal with relativistic mass, at least implicitly.

There is also the concept of "invariant mass", which is the relativistic mass of a system of particles, measured in center-of-mass coordinates so that the total momentum is zero, so that the mass is [math] E/c^2[/math]. This corresponds to the mass that would be measured by a labratory balance for a macroscopic object -- a hot bucket of water in principle weighs more than a cold one. If instead one were to cling to "mass" as being "rest mass" in the macroscopic setting, then the mass measurement of a laboratory balance would not equal the sum of the masses of the molecules of which an object is composed -- creating all sorts of confusion.

My position is to consciously adopt whatever convention is appropriate for the problem at hand, keeping in mind that different authors use different conventions in different situations. "Mass" is not author-invariant.

http://en.wikipedia....cial_relativity

The situation in general relativity is even more murky:

http://en.wikipedia....eral_relativity

http://matheuscmss.w...s-applications/

Can Negative DC voltage be measured? If yes then how and what will be the range of its values.

Yes.

Flip the polarity swithch on your voltmeter, Or just interchange the leads.

The range (in absolute value) is independent of polarity.

 

 

Forgive me for I have sinned in that I do not accept everything about relativity as absolutely true!

You have spoken Ex-Cathedra as a True Believer in the doctrine of relativity. Anyone questioning the dogma is a heretic and a fool. You don't even bother to engage in the conversation enough to address the specifics in my last post. Oh well. I see new developments on the horizon in which good ol' Euclid might find a new respect among relativity theorists, but you will not be among them. Your mind is made up, and you are not about to waste your time with a fool like me. Such fundamentalism right here in science!

One thing that should be abundantly clear is that you are ill-equipped to determine what I think.

My mind is most certainly not made up, nor in my experience is that of any other professional mathematician or scientist. There is no such thing as dogma in science and questioning current theory is the essence of research.

You are correct in only one regard; there is indeed no point in wasting time with a fool.

I think all the arguments with Owl could be resolved with the facts from actual experiments. But here's the problem. We have all kinds of evidence with atomic clocks on airplanes, rockets, and satellites and numerous laboratory experiments that time does indeed slow down with motion, and in the amount predicted by Einstein's special relativity formula. But the evidence for length contraction is indirect. I know of no direct measurement verifying length contaction. I'd be thrilled to learn that I am wrong.

There was an experiment being planned about a year ago, I think it was to be the first direct measurement of length contraction. The trick is to make two sinultaneous position measurements of a rapidly moving object. I have not heard anything further.

Yes, that is the definition I would use. I have seen another (see below)

 

Which does not differ significantly from the above, except it has a slight category-theoretic feel to it: the arrow is the tensor!

 

Let k be a commutative ring and let[math]E_1,E_2,...,E_n[/math] be k-modules.

Consider the class L of all multilinear functions [math]f:E_1 \times E_2 \times ... \times E_n \rightarrow F[/math] where [math]F[/math] varies with [math]f[/math].

The elements may be viewed as the objects of a category. If

[math]f:E_1 \times E_2 \times ... \times E_n \rightarrow F[/math] and [math]g:E_1 \times E_2 \times ... \times E_n \rightarrow G[/math]

are two objects in L, a morphism [math]f \rightarrow g[/math] is a homomorphism [math]h:F \rightarrow G[/math] such that [math] h \circ f = g [/math]

The tensor product of [math]E_1,E_2,...,E_n[/math] denoted [math]E_1 \otimes E_2 \otimes ... \otimes E_n[/math] is a universal repelling object in this category.

It's not like I would ever expect someone to say that QM is a project to deliberately obfuscate theoretical physics and replace it with pure math; and I appreciate the logic that it doesn't make wrong predictions. I'm just more interested in understanding nature than translating it into math so I seek theories that explain things in ways that are comprehensible to my mind. I'm not insisting that nature has to make itself understandable in my terms, as people would accuse me of doing. I just don't think I should have to give up physics because a lot of the math is above me.

"To summarize , I would use the words of Jeans, who said that ‘the Great Architect seems to be a mathematician’. To those who do not know mathematics it is difficult to get across a real feeling as the beauty, the deepest beauty, of nature. C.P. Snow talked about two cultures. I really think that those two cultures separate people who have and people who have not had this experience of understanding mathematics well enough to appreciate nature once." – Richard P. Feynman in The Character of Physical Law

Some people call an element of the tensor product of vector spaces and their dual a tensor. (I probably should have made the tensor product a lot clearer)

 

so [math]T \in V^{*} \otimes V^{*} \cdots \otimes V^{*} \otimes V \otimes V\cdots V[/math] is called a tensor.

 

Some people call the multilinear map [math]V^{*}\times V^{*} \times \cdots V \times V \cdots V \rightarrow R[/math] a tensor.

 

Then you have tensor fields on smooth manifolds that have several ways of defining them. These too are often just called tensors.

 

So, I would say that exactly what you mean by a tensor will depend on the context.

One can always revert to the physicist's definition: A tensor is a symbol having four corners, the two on the right being particularly appealing, to which one may attach an unlimited number of indices.