Money Metric Utility Function: Budget constraints become Utility Functions?

Im currently reading up on the “money metric utility function” (also known as the minimum income function or direct compensation function).

By definition it is defined as:

$$m(text{p},text{x})equiv e(text{p},u(text{x}))$$

Hal Varian writes in Microeconomic analysis (page 109):

it is easy to see that for a fixed $text{x}$, $u(text{x})$ is fixed, so $m(text{p},text{x})$ behaves exactly like an expenditure function: its monotonic, homogenous, concave in $text{p}$, and so on. What is not so obvious is that when when $text{p}$ is fixed, $m(text{p},text{x})$ is in fact a utility function.

The proof is simple: for fixed prices the expenditure function is increasing in the level of utility: if you want a higher utility level, you have to spend more money. In fact, the expenditure function is strictly increasing in $u$ for continuous, local non-satiated preferences.

Hence for fixed $text{p}$, $m(text{p},text{x})$ is a monotonic transform of the utility function therefore itself a utility.

Does this (the bolded statement) mean that we essentially convert our consumer’s budget constraint/budget line into his indifference curves when prices are fixed?

Connectedness: with respect to a larger metric space, or with respect to itself only?

I asked a similar question on the forum earlier, and felt as if I had the issue clarified, but having come upon the problem again in other situations, it is clear that I’m still confused.

Consider the set $X subset mathbb{R} = [a,b] setminus {c}$ for some $c$ such that $a < c < b.$ The textbook says $X$ is disconnected and qualifies its statement by putting forth the “separation” $X = [a,c) sqcup (c,b].$ With respect to the metric subspace $X$, the two intervals $[a,c),(c,b]$ are clopen. However, they are not with respect to $mathbb{R}.$ Do you have to qualify “connectedness” with a “respect to,” as in, $X$ is connected with respect to $mathbb{R},$ but disconnected with respect to itself. Can you comment on the connectedness of the natural numbers?

Every metric on a finite-dimensional vector space is equivalent

I am trying to prove the theorem that states that on every finite-dimensional vector space ($E$ with dimension $n$), every norm is equivalent.
Going through my lecture notes the proof given, starts saying that we can suppose without losing generality that $E =k^n$.

Then it uses the fact that on $k^n$ unit balls are compact and you can easily get the bounds using $||cdot||_infty$ and other generic norm.

My doubt is: why can we asumme $E=k^n$?.

Why having a conserving metric defined on a product space guarantees that projections are isometries?

I wanted to know when the metric (between 2 spaces A and B) is preserved.

Is metrics preserved when isometries are projections that guarantee or define a product space existence within or of product space existence itself ?

Product Space
A Cartesian product equipped with a “product topology” is called a product space (or product topological space, or direct product).

curvature for metric $ds^2 = ydx^2 + xdy^2$

I’m trying to solve problem from Zee’s Einstein gravity in nutshell:
find curvature for metric $ds^2 = ydx^2 + xdy^2$

moreover, in the corresponding chapter curvature is derived from the term of the expansion of the metric near some point : $g_{munu}(x) = g_{munu}(0) + … + B_{munu, rhosigma}x^rho x^sigma + …$
so that curvature is some combination of $B_{munu, rhosigma}$
Is it possible to find curvature in this case without calculating and differentiating christoffel symbols, summing over indices of curvature tensor and etc.?

Constructing vielbein from given metric: example: 2D spherical coordinates

Usual relation between metric and vielbein are given by
begin{align}
g^{munu} = e^{mu}{}_a e^{nu}{}_{b} eta^{ab}
end{align}
where $eta^{ab}$ is flat, $mu,nu$ is curved, ($i.e$, diffeomorphism index : index related with coordinate change) and $a,b$ are (Lorentz index).

I know that for given metric, vielbein form is not unique. Even though i want to construct vielbein and check above and inverse of above independently.

In the usual GR textbook, even though they mention vielbeins, for actual computation they just compute 1-forms (orthonormal basis) and obtain same results.

For example in sphere
begin{align}
ds^2 = dr^2 + r^2 dOmega^2
end{align}
and $e_r = dr$, $e_{theta} = rdtheta $, $e_{varphi} = r sin(theta) dvarphi$, then via cartan’s formalism (via its structure equations), i can compute connection, Riemann tensor and so on.

What i want to know is process of computing vielbein of general given metric. (not necessarily be diagonal)
Before generalizing i want to know some simple case.

Manually Setting Interface metric priority of Network Adapters not preferring lower metric route on Windows

After Connecting to Cisco VPN AnyConnect, Now I have two network interfaces having set same default routes, but with different metric values. Even after manually changing/raising the metric value of one default route(i.e. imposed by VPN from metric value 2 to 1000) to give preference to my default route(metric value 26), it’s still preferring the VPN one(instead of raising the VPN route metric value from 2 to 1000 as you can see in route print output)

Here is my route print output:

route print
===========================================================================
Interface List
 10...90 4c e5 58 9f 09 ......Atheros AR9285 802.11b/g/n WiFi Adapter
 20...00 05 9a 3c 7a 00 ......Cisco AnyConnect Secure Mobility Client Virtual M
                              niport Adapter for Windows
===========================================================================

 IPv4 Route Table
===========================================================================
  Active Routes:
  Network Destination        Netmask          Gateway       Interface  Metric
         0.0.0.0          0.0.0.0      192.168.1.1      192.168.1.2     26
         0.0.0.0          0.0.0.0         10.0.0.1       10.1.105.2   1000
        10.0.0.0        255.0.0.0         On-link        10.1.105.2   1255
      10.1.105.2  255.255.255.255         On-link        10.1.105.2   1255
  10.255.255.255  255.255.255.255         On-link        10.1.105.2   1255
       127.0.0.0        255.0.0.0         On-link         127.0.0.1    306
       127.0.0.1  255.255.255.255         On-link         127.0.0.1    306
 127.255.255.255  255.255.255.255         On-link         127.0.0.1    306
    164.100.28.5  255.255.255.255      192.168.1.1      192.168.1.2     26
 164.100.176.115  255.255.255.255      192.168.1.1      192.168.1.2     26
     192.168.1.0    255.255.255.0         On-link       192.168.1.2    281
     192.168.1.1  255.255.255.255         On-link       192.168.1.2     26
     192.168.1.2  255.255.255.255         On-link       192.168.1.2    281
   192.168.1.255  255.255.255.255         On-link       192.168.1.2    281
       224.0.0.0        240.0.0.0         On-link         127.0.0.1    306
       224.0.0.0        240.0.0.0         On-link       192.168.1.2    281
       224.0.0.0        240.0.0.0         On-link        10.1.105.2   1255
 255.255.255.255  255.255.255.255         On-link         127.0.0.1    306
 255.255.255.255  255.255.255.255         On-link       192.168.1.2    281
 255.255.255.255  255.255.255.255         On-link        10.1.105.2   1255
===========================================================================
Persistent Routes:
Network Address          Netmask  Gateway Address  Metric
      0.0.0.0          0.0.0.0         10.0.0.1     999
===========================================================================

IPv6 Route Table
===========================================================================
Active Routes:
If Metric Network Destination      Gateway
23     58 ::/0                     On-link
 1    306 ::1/128                  On-link
23    306 2001:0:9d38:6abd:348b:29cb:f5fe:96fd/128
                                On-link
23    306 fe80::348b:29cb:f5fe:96fd/128
                                On-link
 1    306 ff00::/8                 On-link
23    306 ff00::/8                 On-link
===========================================================================
Persistent Routes:
None

I doubted of this Persistent route entry block which is there in output above as:

Persistent Routes:
    Network Address         Netmask     Gateway Address  Metric
          0.0.0.0          0.0.0.0         10.0.0.1       999

But after issuing the below delete route command as:

route delete 0.0.0.0 mask 0.0.0.0 192.168.1.1

Now Persistent route entry has gone, its displaying as None.

As you see, Even Metric is 26 which is much lower than 1000, it’s still following 1000 metric route. What is going on?

Network Destination       Netmask          Gateway       Interface  Metric
         0.0.0.0          0.0.0.0      192.168.1.1      192.168.1.2     26
         0.0.0.0          0.0.0.0         10.0.0.1       10.1.105.2   1000 

Even I have done the following:
In my Adapter Settings (Control PanelNetwork and InternetNetwork Connections) Advanced Settings Changed the order of the connections so that my connection priority is top on the list over Cisco AnyConnect VPN.

Still If i tracert google.com, still my traffic going over VPN.
Where is the concept of preferring lower cost matrix?
can someone explain to me, what is going over here?

If someone wants to say Cisco AnyConnect Client is playing here behind the scene,

Shall I no more believe on my route print output’s?
Shall I no more believe on concept of preferring lower metric value over higher ones?

please, I want to have my doubts clear.

Prove the metric space with the set of all real-coefficient polynomials restricted to [0,1] and the…

Prove $(Y,d)$ is separable. To clarify, a function $~f:[0,1]to mathbb{R}$ is an element of $Y$ if and only if $~f(x) = a_0 +dotsc+a_{n}x^{n}$ for some $ninmathbb{N}cup{0}$ and $a_0,dotsc,a_n inmathbb{R}$.

Is this a natural metric on the space of all unoriented lines in a 2D place with positive slope?

Let’s parameterize a 2D (unoriented) line by the slope $m$ and intercept $b$, and let the slope be positive. Thus, we are looking at all lines in the first quadrant. I want to know if there’s a natural way to measure distances between lines or the “size” of a line in the $(m,b)$ space. I know that there’s no natural metric on the space of all possible unoriented lines, but I wonder if there exists one for the restricted case I have described above. Here’s how I am reasoning about it for now.

Two lines with the same slope are different only by the intercept $b$, so that’s trivial. Next, two lines with the same intercept differ by the angle $theta=arctan(m)$. So, am I correct in thinking that the 2D polar coordinate system is the natural coordinate system here? If so, then can I use $b^2+b^2theta^2$ as a metric?

Metric of an Evaporating Black Hole

The famous Hawking calculation is done with an assumption that the background is static, i.e. the evaporation doesn’t change the mass parameter in the metric. Thus, we simply describe the geometry using the static Schwarzschild (or, generically, Kerr-Newman) metric. But clearly, the evaporation actually makes the geometry non-static and thus, the geometry should actually be described using a non-static metric. I am finding a hard time finding out what metric this is.

I think that even if the Hawking calculation is done within the assumption that the background metric is static, one can safely assume that a spherically symmetric radiation will still be being emitted from an evaporating black hole even during the stages where the static assumption is inappropriate. Thus, the natural guess for a non-static metric that describes the geometry of an evaporating black hole would be the Vaidya metric.

But, as discussed in this answer, an outgoing Vaidya metric describes a metric for which the mass parameter is continually decreasing–but this doesn’t describe a black hole geometry, instead, it describes a white hole geometry. Further, as discussed in the same answer, an ingoing Vaidya metric describes a black hole geometry–but with a monotonically increasing mass parameter. Thus, none of the two Vaidya metrics qualify to describe an evaporating black hole.

So, my question is, is there any known metric that can describe a spherically symmetric geometry whose mass parameter decreases with time and the horizon is of the nature that resembles a black hole horizon? If so, then it can be considered as a metric that describes an evaporating black hole.

Edit

I recently read a comment by @JerrySchirmer that the Hawking radiation violates the energy conditions. If so is the case then the argument that an ingoing Vaidya metric has a monotonically increasing mass parameter doesn’t work (as this argument relies on the null energy condition). If someone can provide some canonical references in this regard then it would be truly helpful.