• Quick note - the problem with Youtube videos not embedding on the forum appears to have been fixed, thanks to ZiprHead. If you do still see problems let me know.

General Relativity

My understanding is that proper time in relativity is the elapsed time measured by a clock experiencing two events. So, with respect to the concept of proper time, ΔΤ2 = Δt2 - Δx2 - Δy2 - Δz2, what does it mean for ΔΤ2 to be negative, which can clearly happen for selected values of t,x,y,z? In other words, what is the geometric (or physical) interpretation of negative ΔΤ2 or, equivalently, imaginary values of ΔΤ?
Alternatively the same kind of question question arises when considering ΔS2, from the Minkowski metric, when negative values (or imaginary for ΔS) are derived? What is the geometric or physical interpretation of negative (or imaginary) values of the metric?
Thanks in advance.
 
Sorry, I did not ask the question very well. Clearly, if ΔΤ2 is negative the events cannot be connected by causality. I was interested in learning what the meaning, if any, there may be of a negative value of ΔΤ2? Perhaps I'm looking for some meaning that does not exist outside of causality so my question probably has no real meaning.
 
Perpetual Student said:
Sorry, I did not ask the question very well. Clearly, if ΔΤ2 is negative the events cannot be connected by causality. I was interested in learning what the meaning, if any, there may be of a negative value of ΔΤ2? Perhaps I'm looking for some meaning that does not exist outside of causality so my question probably has no real meaning.
Click to expand...

BowlOfRed's answer is correct - it simply means that the events in question are spacelike separated.

In a Lorentz invariant theory that has implications for causality, in particular that two events at those locations cannot have influenced each other. But it's really a geometric statement - it's like saying two points on earth are separated more in latitude than in longitude.
 
H'ethetheth said:
So I wanted to see what this discussion had gotten up to, and then I saw this sentence.

Funny how a sentence can become completely unintelligible when you don't know what any of the terms mean.
Click to expand...

It puts Star Trek's technobabble in its place, and what's more it's real. No bulldust gets past this crowd.
 
Perpetual Student said:
My understanding is that proper time in relativity is the elapsed time measured by a clock experiencing two events. So, with respect to the concept of proper time, ΔΤ2 = Δt2 - Δx2 - Δy2 - Δz2, what does it mean for ΔΤ2 to be negative, which can clearly happen for selected values of t,x,y,z? In other words, what is the geometric (or physical) interpretation of negative ΔΤ2 or, equivalently, imaginary values of ΔΤ?
Click to expand...
For a clock in inertial motion, proper time is

[latex]\Delta \tau = \sqrt{\left(\Delta t\right)^2 - \frac{\left(\Delta x\right)^2}{c^2} - \frac{\left(\Delta y\right)^2}{c^2} - \frac{\left(\Delta z\right)^2}{c^2}}[/latex]

Make this clock a parallel-mirror light clock, and you soon appreciate that the expression is restricted. There is no physical interpretation for negative proper time. You cannot count a negative number of reflections between the parallel mirrors. Here's something I was talking about on a related matter that might proves useful.
Farsight said:
Take a look at the well-worn expression for a spacetime interval in flat Minkowski spacetime:

[latex]$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$[/latex]

It's related to Pythagoras' theorem, used in the Simple inference of time dilation due to relative velocity. We've got two parallel-mirror light clocks, one in front of me, the other which we've sent with you on an out-and-back trip. I see the light moving like this ǁ in my local clock and like this /\ in the moving clock. Treat one side of the angled path as a right-angled triangle, and the hypotenuse is the lightpath where c=1 in natural units, the base is the speed v as a fraction of c, and the height gives the Lorentz factor γ = 1/√(1-v²/c²). Notice there's no literal time flowing in these clocks, merely light moving at a uniform rate through the space of our SR universe. From which we plot straight worldlines through the abstract mathematical space we call Minkowski spacetime. Also note that the underlying reality behind the invariant spacetime interval between the start and end events is that the two light-path lengths are the same. Macroscopic motion comes at the cost of a reduced local rate of motion, because the total rate of motion is c. Hence the minus in front of the t. It really is that simple.
Click to expand...
 
Is there any particular history or reason [latex]T_\mu_\nu[/latex] is called the stress-energy tensor? I notice that it is also called the energy-momentum tensor, which is consistent with the fact that its components relate to the energy and momentum of any existing matter/energy and the resultant gravitational field. I have also seen "stress–energy–momentum tensor." A related question is, where does the concept of stress come in?
Thanks in advance.
 
What does Maxwell say about stress?
Maxwell (Matter and Motion) said:
37. MUTUAL ACTION BETWEEN Two BODIES STRESS
The mutual action between two portions of matter receives different names according to the aspect under which it is studied, and this aspect depends on the extent of the material system which forms the subject of our attention.

If we take into account the whole phenomenon of the action between the two portions of matter, we call it Stress. This stress, according to the mode in which it acts, may be described as Attraction, Repulsion, Tension, Pressure, Shearing stress, Torsion, etc.
Click to expand...
Stress is an important part of a complete description of a collection of matter. It is an important part of the description of the energy of that collection. As such, it is a potential influence on the geometry of spacetime and is part of the stress-energy tensor.
 
Perpetual Student said:
Is there any particular history or reason [latex]T_\mu_\nu[/latex] is called the stress-energy tensor? I notice that it is also called the energy-momentum tensor, which is consistent with the fact that its components relate to the energy and momentum of any existing matter/energy and the resultant gravitational field. I have also seen "stress–energy–momentum tensor." A related question is, where does the concept of stress come in?
Thanks in advance.
Click to expand...
As I recall, Tμν can be interpreted as the flow of μ-momentum density across a hypersurface of constant ν-coordinate. I haven't thought about this in a long time, but I think it means that T00 is the energy density of matter, T11, T22 and T33 are momentum densities, and the (nine) Tij with i and j in the set {1,2,3} are stresses.
 
Fredrik said:
As I recall, Tμν can be interpreted as the flow of μ-momentum density across a hypersurface of constant ν-coordinate. I haven't thought about this in a long time, but I think it means that T00 is the energy density of matter, T11, T22 and T33 are momentum densities, and the (nine) Tij with i and j in the set {1,2,3} are stresses.
Click to expand...

Getting a handle on this energy-momentum tensor is a challenge. So, does it follow that these stresses are shear-like in nature if the [latex] i \neq j [/latex] components are not zero?
 
Perpetual Student said:
Getting a handle on this energy-momentum tensor is a challenge. So, does it follow that these stresses are shear-like in nature if the [latex] i \neq j [/latex] components are not zero?
Click to expand...
I don't really know, but I looked at the Wikipedia article on shear stress, and it looks like each of the Tij (with i≠j and i,j in {1,2,3}) represents what they call a "normal" stress. However, if there are non-zero normal stresses in several different directions at one point, doesn't that mean that there's a shear stress? It seems that way to me, but I haven't thought about it enough to be sure. I wouldn't be surprised if it turns out that I'm wrong. I have never really studied any form of stress.
 
Perpetual Student said:
Is there any particular history or reason [latex]T_\mu_\nu[/latex] is called the stress-energy tensor? I notice that it is also called the energy-momentum tensor, which is consistent with the fact that its components relate to the energy and momentum of any existing matter/energy and the resultant gravitational field. I have also seen "stress–energy–momentum tensor." A related question is, where does the concept of stress come in?
Click to expand...
See this wikiversity article which says "According to Simmonds, "the name tensor comes from elasticity theory where in a loaded elastic body the stress tensor acting on a unit vector normal to a plane through a point delivers the tension (i.e., the force per unit area) acting across the plane at that point."
 
Think of elasticity when you read up on Einstein Field Equations. That shear stress is in space. Somebody mentioned Maxwell, have a google on Maxwell and elastic. You don't find much reference to "elastic space" in the textbooks these days, but that's what it comes down to. What's a bit of a pity when it comes to general relativity, is that the bowling-ball analogy is overused. A better analogy for a gravitational field is a block of jelly where you inject more jelly in the centre, creating a spherical pressure gradient. A ripple in the jelly veers when it encounters this pressure gradient. NB: both pressure and stress are measured in Pascals, pressure is essentially stress without a direction to it. See this stress and pressure page by Philip Candela.
 
Thanks, that's a good link, but I've become weary of analogies like jelly, bowling balls, rubber sheets, etc. without the actual mathematics. At this stage, for my needs, only the mathematics will provide the kind of information and understanding I am interested in. It will take some time, but I'm fortunate to have the time and the learning tools readily available today are amazingly powerful: Internet lectures, text references, papers, forums, etc. I'm finding that tensors and differential geometry are not quite as formidable as I feared.
I only regret not having started a few years ago.
 
No probs. If I can offer a word of advice: when you're studying the mathematics, always make sure that you know what each term relates to wherein a concentration of energy affects the surrounding space altering the motion of a light wave moving through it.
 
Last edited:
Farsight said:
No probs. If I can offer a word of advice: when you're studying the mathematics, always make sure that you know what each term relates to wherein a concentration of energy affects the surrounding space altering the motion of a light wave moving through it.
Click to expand...
So you have mastered the mathematics of tensors? Why don't you ever answer a mathematical question?
 
I wouldn't say I've mastered it, but it isn't rocket science, you start with vectors then move up to matrices. A tensor is a matrix. And I have answered mathematical questions, see post #387. Sure I focus on the physic discussion and sometimes complain that mathematics can distract from that, but I'm not some guy who shirks all mathematics. Now stop carping, stay on topic, and respect Perpetual Student's thread. Try to make a useful contribution to it, and try not to derail it.
 
You must log in or register to reply here.

Back
Top Bottom