The discussion with Will Happer about the pyrgeometer recorded in recent posts illuminates the distinction between state and and process in physics, with process connecting to transition between states. For the pyrgeometer the question is if it primarily/directly measures (a) temperature/state or (b) radiation/process. Happer says (b) and and I say (a), while a pyrgeometer manual states that (b) is computed/derived from (a) and so is not directly measured.
To get perspective, consider a 3d coordinate system by which we can measure/record the position of an object as a form of state, assuming it does not change position. This seems like a fairly straight forward thing to do.
Assume now that the body changes position with time, which brings in the notion of time to which we will return, and we then meet the concept of velocity as change of position vs change of time. It then seems natural to view velocity as a process variable since it involves transition between two different positions/states over some (infinitesimally small) change of time. If position is easy to measure, velocity seems to be more difficult since it involves measuring a small change of position over a small change of time.
So it seems natural to view position as a state variable and velocity as a process variable, with a natural connection of transition to change of time. Translated to the pyrgeometer it suggests that (a) is easy (state) and (b) difficult (process). Accordingly a pyrgeometer can be expected to be designed to measure (a). If simple works, why aim for difficult?
Similarly, conductive heat flux scaling with temperature gradient is a process variable analogous to radiative heat flux/radiation.
This brings up the question if there is any difference between space variable(s) and time variable? Yes, if it is natural to view position as a state variable, then it may be natural to view time as a process variable as something always in transition form one time instance to the next.
Thus there seems to be a clear distinction between space (state) and time (process) variable. If so, Einstein's key notion of space-time variable with time acting like a 4th space dimension does not seem to be natural, since it makes time a state variable instead of process variable and if you mess things up then you will have a mess.
Inlägg
You are correct as regards the pyrgeometer... it measures the temperature differential and mathematically derives the radiant flux. That it does so via a misuse of the S-B equation (using the form of the S-B equation meant for idealized blackbody objects and thus assuming emission to 0 K, which inflates radiant exitance of all objects and thus invents a wholly-fictive 'cooler to warmer' backradiation energy flow) is relevant here... I'm not sure what the process variable would be in this case, given that it presupposes emission to 0 K... if it did it correctly, the process variable would be the ambient temperature or energy density (that which they've pinned to 0 K in current pyrgeometers).
SvaraRaderaSpace-time combines the three DOF of space and the one DOF of time into a single 4-dimensional manifold.
Thus an object can be said to be at point [x,y,z] at time [t].
This makes sense, as it takes time to traverse space... an object cannot instantaneously jump from one space-time point to another.
Now, Special Relativity is predicated upon the speed of light being a constant, but General Relativity adds a few caveats to that... it is constant from one point in space-time to another if both points are en vacuo and both have the same gravitational acceleration (as well as the path between the two points having the same gravitational acceleration). Einstein reiterated this point several times.
1913: “I arrived at the result that the velocity of light is not to be regarded as independent of the gravitational potential. Thus the principle of the constancy of the velocity of light is incompatible with the equivalence hypothesis.”
The Equivalence Principle states that there is no difference between gravitational acceleration and translational acceleration via other-than-gravitational means... both are acceleration.
1916: “In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity.”
1920: “Second, this consequence shows that the law of the constancy of the speed of light no longer holds, according to the general theory of relativity, in spaces that have gravitational fields. As a simple geometric consideration shows, the curvature of light rays occurs only in spaces where the speed of light is spatially variable.”
This is how gravitational lensing (ie: the curvature of light rays around a gravitating body) occurs, which was what originally corroborated Einstein’s theory.
And given that time is also relative, this necessitates that an observer synchronize clocks between different frames according to their own frame. Even optical clocks at different altitudes on Earth cannot keep their time synchronized, time passes more slowly the deeper you are in a gravity well or the faster you're traveling.
T_1 = T_0/(√(1 − v^2 c^2)) …faster object, slower time
T_1 = T_0/(√(1 − (2GM/c^2 R))) …farther from a gravity well, faster time
That's why GPS satellites have to synchronize their clocks to 'surface time'. If allowed to time-drift for a single day without correction, the GPS satellites would be off by ~38000 nanosec, and given the speed of the GPS satellites, that means GPS would be off by ~15 cm / day (speed of satellite * 38 µsec/day). In actuality, the GPS clocks are synchronized weekly, so the maximum error due to relativistic effects is ~105 cm.
Now, a state variable is one which is used to mathematically describe the state of a dynamical system, whereas the process variable is that variable which is monitored or controlled to determine the change in state of a dynamical system. A process variable is a state variable... it's the monitored or controlled state variable. It still describes some portion of the state of the system, but it's the control variable by which evolution of the system is delineated.
So the space-time manifold can be used such that: space is the state and the process variable, that space is the state variable and time is the process variable, or that time is the state variable and space is the process variable.
SvaraRaderaThis relativity plays all sorts of havoc with simultaneity of events:
https://upload.wikimedia.org/wikipedia/commons/7/78/Relativity_of_Simultaneity_Animation.gif
Three-dimensional space defines the distance between two points (via the Pythagorean Theorem) thusly:
(Δd)^2 = (Δx)^2 + (Δy)^2 + (Δz)^2
You’ll note that doesn’t account for time, which the spacetime interval equation does thusly:
(Δs)^2 = (cΔt)^2 – (Δx)^2 – (Δy)^2 – (Δz)^2
If (Δs)^2 > 0, the spacetime interval is timelike, meaning two events are separated by more time than space
If (Δs)^2 < 0, the spacetime interval is spacelike, meaning two events are separated by more space than time.
If (Δs)^2 = 0, the spacetime interval of two events on a worldline of something moving at c is zero. We know this is true because we know photons (traveling at c in vacuum) do not experience time and thus do not change with distance (sans external influence). Thus when we look out into space, we’re literally looking back in time. The photons reaching us carry the information from the time they were emitted.
In fact, if one measures distance in light-seconds rather than meters, one can use:
(Δs)^2 = (Δx)^2 + (Δy)^2 + (Δz)^2 – (Δt)^2