كيف يؤثر حجم الثقب الأسود العاري على كرة الفوتون الخاصة به؟

لإثبات المعادلة. (4.36) ، فإن أداتنا هي طريقة متغيرة تتحد مع مضاعفات لاغرانج. الهدف الوظيفي الآن هو

وفقًا للاستنتاج الوارد في الملحق C ، يمكن تحويل ذلك إلى مهمة إيجاد سطح حرج بين الأسطح الموضوعة على ورقة فارغة (u = 0 ). على غرار (4.16) ، لدينا العلاقة التالية

كما نريد إيجاد الحد الأعلى لـ (A_S ) عندما س هو كرة فوتونية ، هناك عدد قليل من القيود على السطح س.

أولاً ، على قدم المساواة-ر surface, we have know that a photon sphere is a critical surface which makes the integration (int phi ^<-2> ext S) to be extremal. In the null sheet (u=0) , Appendix C shows that a photon sphere should also be a critical surface which makes the integration (int phi ^<-2> ext S) to be extremal on the null sheet (u=0) . On the null sheet (u=0) , we have (int phi ^<-2> ext S=int r_S^3e^<-4eta >/(Ve^<-2eta >) ext Omega ) . The extreme condition means that س on the null sheet (u=0) should satisfy following equation

As we now fix the value of functional (>=>_0) , we have following constraint

Thus, we construct following target functional on the null sheet (u=0) with two Lagrangian multipliers

Here () are two Lagrangian multipliers and (lambda _2) is constant. The critical values of (A_>) in equal-ر surface are given by the critical values of F in the null sheet (u=0) .

To solve the variational problem, we separate functional F into two parts

The (partial _r(Ve^<-2eta >)) and (partial _reta ) in (F_>) are given by Eq. (4.9). The (Ve^<-2eta >) in (F_>) is given by Eq. (4.18). Using Eqs. (4.9), (4.10) and (4.18) we find

Here (eta _S=eta |_S) , (N_<1S>^2) and (N_<2S>^2) are defined by Eq. (4.22) but restricted on surface س. We use the lower index “س” to explicitly show that they are surface quantities. Then using the conformal transformation (4.25), we have

Functional (F_>) only involves quantities at the surface س, but functional (F_>) involves quantities at surface س as well as in bulk region between س and (> ). The functional F now depends on 6 surface variables (,N_<2S>,eta _S, Psi _S, W_S, r_S>) and 4 bulk variables (). We can treat all these variables as independent variables.

The (Psi _S, W_S) , (N_<1S>) and (N_<2S>) appear only in boundary part (F_> ). Using Eq. (4.46), we find that the variations with respective to (W_S, N_<1S>) and (N_<2S>) show

Then variation of (Psi _S) reads

This shows that (lambda _1(y^A)) is a constant. The variation with respective to (eta _S) involves both (F_>) and (F_> ). The result reads,

As (lambda _1) is constant, we integrate Eq. (4.49) on س and find

As (V>0) , we find (lambda _2=0) , which implies (>>^2eta _S=0) Footnote 3 and so (eta _S) is constant.

The variables () appear only in bulk part (F_> ). The variation with respective to (N_1) and (N_2) shows (N_1=N_2=0) . Now we have

Then we variation with respective to (Psi ) shows (>>^2[Theta (r_S-r)r_S^<-1>]=0) and so (r_S) is constant. The variation of (eta ) shows (>>^2eta =0) and so (eta =eta (r)) when (r<r_h<r_S) . Thus, we have following results at the surface س

Note that (Psi _S=0Rightarrow <^<(2)>R>|_S=2) . Take these into Eqs. (4.38) and (4.40) we have

Thus, we find that (F|_>=e^<4eta _S>/(3>_0)) . The on-shell values of target functional are not unique. ثم لدينا

This proves Eq. (4.36) and so we obtain (A_>le A_>/3) . It needs to note that above proof is still true for every connected photon spheres, i.e. (A_,i>le A_,i>/3) . Thus, we prove one part of our stronger conjecture (3.11).

Proof of (A_>/3le 36pi M^2)

The solution of Eq. (4.10) can be written as

In the gauge that horizon has constant (r=r_h) , we find

This shows the Penrose inequality (9A_<>>/4le 36pi M^2) .

To show (A_>/3le 36pi M^2) , we immerse the photon sphere into the null sheet (u=0) and take the gauge that the radius of outmost photon sphere (r_<>>) is constant. The size of shadow reads

In general, the horizon radius (r_h) is no longer constant. Eq. (4.56) becomes

In following we will use variational method to show that, in all black holes of which the size of shadow is (A_>=12pi a_0^2) , the minimal value of mass is (a_0/3) . The main idea is as follow.

We treat م as a functional, the positive mass theorem shows (Mge 0) . Thus, the functional م is bounded from below. Then the minimum value can be found according to variational problem. As (>_>) is a photon sphere, so constraint (4.38) should also satisfied. We construct the target functional

Here (lambda _1(x^A)) and (lambda _2) are two Lagrangian multipliers and (lambda _2) is constant. The (lambda _2) insures that the size of shadow is fixed to be (12pi a_0^2) and the (lambda _1) insures that condition (4.38) is satisfied.

We use the variable transformations (4.22) and find

The functional (>) contains the boundary part and bulk part. In principle, we can follow the similar step of Sect. 4.3 to solve the variational problem. However, in this case, we have simpler method.

We first choose (eta , N_1, N_2) and (K:=sqrt>) as four independent bulk variables. The bulk variations with respective to (eta , N_1) and (N_2) only involves the third line of Eq. (4.604.61). The result shows

N_1=N_2=0 Rightarrow partial _rh_=0,. نهاية$

Thus, the on-shell geometry outside photon sphere (>_>) becomes spherically symmetric. We then find that

In spherically symmetric case, we have shown that (min M=sqrt>/(108pi )>=a_0/3) . Thus Eq. (4.62) shows (min M=a_0/3) . Then we finish the proof of (A_>/3le 36pi M^2) in the case without spherical symmetry. Note that above proof also implies that (A_>/3=36pi M^2) is true only if the geometry outside (>_>) is Schwarzschild.

Now we make a short summary on this section. We use variational method to prove (9A_<>>/4le A_>/3) and (A_>le A_>/3le 36pi M^2) under two assumptions: (1) weak and strong energy condition are satisfied, and (2) outmost horizon and outmost photon sphere are connected and smooth. Our proof also implies a rigidity theorem: (A_>=36pi M^2) only if the geometry outside outmost photon sphere is Schwarzschild. Our proof on (A_>le A_>/3) is also true when outmost photon sphere is not connected. As the result, we obtain (A_,i>le A_,i>/3) for every connected branch and so prove a part of our stronger conjecture (3.11). The lower bound (9A_<>>/4le A_>) in spherically symmetric case involves some non-typical energy condition (2.15). It is not clear how to generalize it into the case without spherical symmetry. We leave the study of this inequality for the future.

Comment on smoothness: In above proofs, we assumed that the outmost photon sphere (>_>) is smooth. This requirement can be be replaced by piecewise smoothness. The reason is that, in variational method, the target functionals and constraints are defined in integrations. For a outmost photon sphere (>_>^<(i)>) , we can always find a smooth surface (>_,varepsilon >^<(i)>) which satisfies: (1) (forall varepsilon >0) , the maximal volume enclosed by (>_>^<(i)>) and (>_,varepsilon >^<(i)>) is smaller than (varepsilon M^3) , i.e.

أين الخامس is arbitrary spacelike 3-dimensional surface and satisfies (partial V=>_>^<(i)>cup >_,varepsilon >^<(i)>) , and (2) (A_,i>) and (A_,i>) are approximated in arbitrary accuracy

and (3) the constraint (4.38) is broken arbitrarily small

We can use the smooth surface (>_,varepsilon >^<(i)>) to replace (>_>^<(i)>) and obtain the same conclusion. Because of this reason, the conclusions in our above proofs are still valid if assume the outmost photon sphere is piecewise smooth.

6 Answers 6

I am neglecting the effect of tidal forces on the integrity of any object

Well, don't be doing that! Those forces can be used to answer your question.

Given a couple of bars of known length and mass, and a black hole of known mass, you could measure the relative differences in length of a bar placed parallel to the local gravity vector and one placed perpendicular to it using eg. interferometry. The vertical bar will be stretched by tidal forces the closer you get, and the amount of stretch will be predictable and related to your distance from the hole.

As an alternative that uses boring old orbital mechanics instead of exciting space-mangling super-intense gravity fields, you could measure the period of your orbit by observing background stars, which you can throw into the usual equations to work out the orbital radius. This works when you are far enough away from the hole to avoid frame dragging effects, but as you get closer, more clever mathematics will be required.

Determine the location of the photon sphere using laser beams.

The photon sphere is located farther from the center of a black hole than the event horizon. Within a photon sphere, it is possible to imagine a photon that's emitted from the back of one's head, orbiting the black hole, only then to be intercepted by the person's eyes, allowing one to see the back of the head.

You here use the photon sphere as a proxy for the event horizon. You will swing your (shark-mounted) laser slowly from left to right such that it will trace out a diameter of the circular region of interest. There will be a point where you can detect your photons coming back to (your shark) as they loop around the black hole. That distance from the center is just outside the photon sphere which is just outside the event horizon. As your laser continues to the right its photons will be lost into the black hole. As it emerges on the left your will detect the photons coming around from the right. You have now marked the lateral edges of the photon sphere here used as a proxy for the black hole. Should you suspect the region is other than circular you can draw a few more diameters with your laser.

You do not need to get close to do this. You can do this from some distance, which I recommend at least the first few times.

Datalink frequency shift.

The probe is in communication with a relay station live streaming the event back to NASA and eager nerds like me. The communication is bidirectional, and the probe is constantly acknowledging packets received. (Think TCP protocol but tweaked for space.)

The constant stream back of acknowledgements keeps the communication channel constantly active, allowing the probe to measure the frequency shift of the data link.

Because it knows that the relay is far enough away from the black hole (and traveling slowly enough) that time dilation isn't an issue to N significant digits, it can calculate the exact time dilation from the frequency shift of the known communication frequency.

Eg the 5Ghz radio signal is arriving at 25.453Ghz - so that relay stations clocks are running 5.0906 times faster than mine. That's means my clocks are running at 19.644% of real time. You now know your time dilation factor to as accurately as your communication unit can measure the changing frequency.

Assuming you're not orbiting at relativistic speeds, from this, you can calculate your approximate distance from the black hole.

If you are orbiting at relativistic speeds, ie, you're getting in that close, you can measure the change in time dilation (via frequency shift) at regular intervals (in probe time). I'm not smart enough to be 100% certain this will work without coding a simulation, but my gut instinct is that there will only be one valid orbital ellipse possible for a given sequence of time dilation measurements.

• time dilation from the gravity of the blackhole and the probes distance to it.
• time dilation from the speed of the probe.
• doppler shifting based on the relative velocity of the probe and the relay.

That sequence of changing times should give only one ellipse and your phase on that ellipse. From that, you can do your orbital maneuvers.

Hello PG, !

There's a ton of stuff waiting to be looked at if you google 'picture of a black hole'

I think we had a thread on this one in PF. It's beautiful, but the explanation isn't for the faint-hearted .

Veritasium had a pretty good description of the images from earlier this year and why they look that way.

Thanks both! I have seen the image of that hole and I sorta get why it looks like that.

But lets imagine an ancient bh without a disk and me being an observer outside its radius, would the gravitational lensing cause the stars etc behind it to "appear in front of the hole"(Effectively making it indiscernable)? Or would there be some "artifacts" from the lensing that would make it "visible"?

Iam sorry if I make it confusing, having a hard time posing it properly (english isnt my 1st or 2nd language)

They're not discussing "what is the case" as far as the answer to your question: they both agree on this:

Which means the answer to your original question is "yes". They are only discussing the exact optical size of the black disk.

Indeed - I don't think I'm disagreeing with Orodruin's answer except for the angular size of the black area. And we probably disagree on language rather than physics.

There are a few plots of light rays near black holes near the end of this thread, which may be of interest.

Thank you very much, that is very helpfull indeed.

ps. (very long)
Please understand not eveyone is in a position to discuss these things on a campus, with friend or at work. I work at a chemical plant as a shift supervisor and there are plenty of engineers and chemical engineers to talk about every day physics/chem. I have a family and some friends, none of whom are even remotely interested in these things. I have followed alot of online material from lectures to "mainstream physics" like CBSst.
Point is: There is ALOT of material out there, but unless there are people willing to explain/go into some nuance and have a discussion it is very very difficult to self-educate oneself in these topics. Sometimes all one needs is just a little nudge.
I apologize for my crossness, but you dont know half of how lucky you guys are living/studying these field in an enviroment of likeminded people. I am not that lucky, and hoped to find some here. That all.

And indeed you will. No one, myself included, intended any rudeness towards you. You are probably more accustomed to traditional social media where insults and flames are rampant and it is sometimes reasonable to infer that rudeness is intended. We work pretty hard here to NOT be rude to individuals, but ideas are another thing entirely. This next does not apply to you, I'm just making you aware: when ideas that don't make sense crop up, folks are quick to point out the flaws, but the intent is never to rude to the poster, just to lead them to understanding. Likewise, and this does apply to you, when people seem to have made little or no effort on their own, this is often pointed out. Again, no rudeness is intended.

So, I still mean what I said about doing some research on your own. You certainly do raise the good point that there is just so MUCH stuff out there that it can be difficult. Still the effort should be made.

I, like you, have absolutely no one that I talk to in person about science or math. I took college physics over 50 years ago and only a few years back decided to start studying cosmology and quantum mechanics, at the amateur level. I watched TONS of TV shows and read LOTS of pop-science books and then discovered this place and found out that a large amount of what I "knew" was totally wrong because pop-sci presentation are entertainment, not education. HERE, I actually learned a bit of physics and you will to if you stick around.

tnx both! Well I was just a little frustrated (and acted that out abit too much :) ) to be pointed back to the only place I had, and one that I can never have a discussion with. I am happy too be here

@BvU we make metal alkyls, not many of us around, pretty interesting stuff with some nice challenges safety wise.

Reissner-Nordström Black Holes

Assume the existence of two commuting Killing vectors – timelike ξ α and axial η α (ξ α ξ_α <0,η α η_α> 0),ξ α normalized at (asymptotically flat) infinity, η α at the rotation axis. They generate two-dimensional orbits of the group جي₂. Assume there exist 2-spaces orthogonal to these orbits. This is true in vacuum and also in case of electromagnetic fields or perfect fluids whose 4-current or 4-velocity lies in the surfaces of transitivity of جي₂ (e.g., toroidal magnetic fields are excluded). The metric can then be written in Weyl’s coordinates (t,ρ, φ, z)

The most celebrated vacuum solution of the form [12] is the Kerr metric for which يو, ك, أ are ratios of simple polynomials in spheroidal coordinates (simply related to (ρ,z)). The Kerr solution is characterized by mass م and specific angular momentum أ. ل أ 2 > م 2 , it describes an asymptotically flat spacetime with a naked singularity. ل أ 2 ≤ م 2 , it represents a rotating black hole that has two horizons which coalesce into a degenerate horizon for أ 2 = م 2 – an extreme Kerr black hole. The two horizons are located at ص_± = م ± (م 2 − أ 2 ) 1/2 (ص being the Boyer–Lindquist coordinate (يرى Stationary Black Holes )). As with the Reissner–Nordström black hole, the singularity inside is timelike and the inner horizon is an (unstable) Cauchy horizon. The analytic extension of the Kerr metric resembles الشكل 2 (see Frolov and Novikov (1998), Hawking and Ellis (1973), Misner وآخرون. (1973), Ortín (2004), Semerák وآخرون. (2002), Stephani وآخرون. (2003), and Wald (1984) for details).

Thanks to the black hole uniqueness theorems (يرى Stationary Black Holes ), the Kerr metric is the unique solution describing all rotating black holes in vacuum. If the cosmic censorship conjecture holds, Kerr black holes represent the end states of gravitational collapse of astronomical objects with supercritical masses. According to prevalent views, they reside in the nuclei of most galaxies. Unlike with a spherical collapse, there are no exact solutions available which would represent the formation of a Kerr black hole. However, starting from metric [12] and identifying, for example, ض = b = const. و ض = −b (with the region −b & lt ض & lt b being cut off), one can construct thin material disks which are physically plausible and can be the sources of the Kerr metric even for أ 2 > م 2 (see Bičák (2000) for details).

In a general case of metric [12] , EEs in vacuum imply the “Ernst equation” for a complex function F من ρ و ض:

or, equivalently, ( R f ) Δ f = ( ∇ f ) 2 , where F = e 2يو + ib, يو enters [12] , and b(ρ, ض) is a “potential” for أ(ρ, ض): أ_,ρ = ρe −4يو b_,ض, أ_,ض = − ρe −4يو b_,ρ ك(ρ, ض) in [12] can be determined from يو و b by quadratures. Tomimatsu and Sato (TS) exploited symmetries of [13] to construct metrics generalizing the Kerr metric. Replacing F بواسطة ξ = (1 − F)/(1 + F), one finds that in case of the Kerr metric ξ −1 is a linear function in the prolate spheroidal coordinates, whereas for TS solutions ξ is a quotient of higher-order polynomials. A number of other solutions of eqn [13] were found but they are of lower significance than the Kerr solution (cf. Stephani وآخرون. (2003) , Chapter 20).

These solutions inspired “solution-generating methods” in general relativity. The Ernst equation can be regarded as the integrability condition of a system of linear differential equations. The problem of solving such a system can be reformulated as the Riemann–Hilbert problem in complex function theory (يرى Riemann–Hilbert Problem and Integrable Systems: Overview ). We refer to Stephani وآخرون. (2003) and Belinski and Verdaguer (2001) where these techniques using Bäcklund transformations, inverse-scattering method, etc., are also applied in the nonstationary context of two spacelike Killing vectors (waves, cosmology). In the stationary case, all asymptotically flat, stationary, axisymmetric vacuum solutions can, in principle, be generated. It is known how to generate fields with given values of multipole moments, though the required calculations are staggering. By solving the Riemann–Hilbert problem with appropriate boundary data, Neugebauer and Meinel constructed the exact solution representing a rigidly rotating thin disk of dust (cf. Stephani وآخرون. (2003) and Bičák (2000) ).

A subclass of metrics [12] is formed by static Weyl solutions with أ = b = 0. Equation [13] then becomes the Laplace equation Δيو = 0. The nonlinearity of EEs enters only the equations for k : k , ρ = ρ U , ρ 2 − U , z 2 , k , z = 2 ρ U , ρ U , z . The class contains some explicit solutions of interest: the “linear superposition” of collinear particles with string-like singularities between them which keep the system in static equilibrium solutions representing external fields of counter-rotating disks, for example, those which are “inspired” by galactic Newtonian potentials disks around black holes and some other special solutions ( Stephani وآخرون. 2003, Bonnor 1992, Bičák 2000, Semerák وآخرون. 2002) .

There are solutions of the Einstein–Maxwell equations representing external fields of masses endowed with electric charges, magnetic dipole moments, etc. ( Stephani وآخرون. 2003 ). Best known is the Kerr–Newman metric characterized by parameters م, أ, and charge س. ل م 2 ≥ أ 2 + س 2 it describes a charged, rotating black hole. Owing to the rotation, the charged black hole produces also a magnetic field of a dipole type. All the black hole solutions can be generalized to include a nonvanishing Λ (for various applications, see Semerák وآخرون. 2002 )). Other generalizations incorporate the so-called Newman–Unti–Tamburino (NUT) parameter (corresponding to a “gravomagnetic monopole”) or an “external” magnetic/electric field or a parameter leading to “uniform” acceleration (see Stephani وآخرون. (2003) and Bičák (2000) ). Much interest has recently been paid to black hole (and other) solutions with various types of gauge fields and to multidimensional solutions. References Frolov and Novikov (1998) and Ortín (2004) are two examples of good reviews.

Shadows of rotating black holes

In order to make this article self-sufficient, we will briefly introduce some basics in this section. The detailed information can be found in Refs. [52, 53].

In astrometry, the angle ( epsilon ) between two incident light rays can be expressed by the following formula [57]:

Here, ك و ث are tangent vectors of the two light rays, respectively. ( gamma ^ <*>) is the projection operator, ( gamma _< u >^=delta _< u >^+u^ u_ < u >) , for a given observer, whose 4-velocity is denoted by vector ش.

Sketch of measuring the shadow of a spherically symmetric black hole. أ The observers are located at ( heta =0) , and ث is a null geodesic. The angle (psi ) is the angular radius of shadow. b The observers are located at ( heta =pi /2) . ك, ث، و ل are light rays from photon region. (alpha ) , (eta ) and (gamma ) are angles between ك و ل, ل و ث, ك و ل respectively

Generally speaking, the metric of a rotating black hole can be written as

The 4-velocity of a static observer is ( u=frac<1>>> partial _ ). For the asymptotically de Sitter spacetime, there is a cosmological horizon. The observers are fixed at the domain of outer communication that is the region between the event horizon and the cosmological horizon [38]. Figure 2 is a schematic diagram of observers located at ( heta =0) and ( heta =pi /2) respectively. When the observers located at ( heta =0 ) , they will find that the shadow is a disk and the angular radius is

Here, we have choose a light ray ( l=left( l^<0>,l^<1>,l^<2>,l^<3> ight) ) comes from the photon region which is filled by spherical null geodesics [50]. “ ( ext ) ” represents the sign function. For observers located at ( heta >0 ) , the shadow’s silhouette is not a perfect circle as a consequence of the frame dragging effect. As an example, assume the observer located at ( heta =pi /2 ) . Let ( k=left( k^<0>,k^<1>,0,k^<3> ight) ) represent a light ray from a prograde orbit which moves in the same direction as the black hole’s rotation, and ( w=left( w^<0>,w^<1>,0,w^<3> ight) ) represent a light ray from a retrograde orbit that moving against the black hole’s rotation. As mentioned in Ref. [58], a prograde photon orbits the black hole at a smaller radius than that of a retrograde photon because of the well-known Lense–Thirring effect. One can get the angle of the two light rays, in such a way that

where (mathcal equiv k^<3>/k^<1>,mathcal equiv w^<3>/w^ <1>) , and ( hbox (k, w)=< ext >left( cos gamma ight) =hbox left( g_<11>+left( g_<33>-g_<03>^<2>/g_<00> ight) mathcal mathcal ight) ) .

Similarly, the angle ( alpha ) between a light ray ( l=left( l^<0>,l^<1>,l^<2>,l^<3> ight) ) from the photon region and ك is

and the angle ( eta ) between the light ray ل و ث is

In above equations, ( mathcal _ <2>equiv l^<2>/l^<1>, mathcal _ <3>equiv l^<3>/l^ <1>) , ( hbox (k, l)=< ext >left( cos alpha ight) =hbox left( g_<11>+left( g_<33>-g_<03>^<2>/g_<00> ight) mathcal mathcal _<3> ight) ) , and ( hbox (w, l)=< ext >left( cos eta ight) =hbox left( g_<11>+left( g_<33>-g_<03>^<2>/g_<00> ight) mathcal mathcal _ <3> ight) ) . Of course, these rays all come from the photon region.

The angles ( gamma ) , ( alpha ) and ( eta ) can provide us the shadow of black hole in the celestial sphere like Fig. 3. One can imagine that an observer at the center of sphere receives light rays from the photon region in Fig. 3. Thus, the tangents of light rays ك و ث are along (mathrm) and (mathrm) respectively, and their included angle is (gamma =angle mathrm). Similarly, the light ray ل is along (mathrm) and (alpha =angle mathrm) , (eta =angle mathrm). We can use (gamma ) as a representation of the size of shadow, that is, the larger (gamma ) , the larger the size of shadow.

For the sake of convincing in researching the shadow, one can use the following stereographic projection (Fig. 4) for the celestial coordinates to describe the shape of shadow in a 2د-plane [52].

Here, ( Phi equiv angle mathrm ) and ( Psi equiv pi /2-angle mathrm ) are azimuth angle and polar angle in celestial coordinate system.

In order to quantitatively describe shadow’s shape, a distortion parameter ( Xi ) in terms of ( alpha ) , ( eta ) and ( gamma ) is introduced, which is defined as

where ( Xi ) ranges from 0 to ( pi ) . We only care about the hemisphere where the shadow is. If the shadow is not deformed, the boundary is a perfect circle, which means that the shadow is a spherical cap on the hemisphere. In this case, ( cos Xi =0 ) . If the shadow is deformed, (cos Xi e 0) . Specifically, the symbol of ( cos Xi ) corresponds to the following two cases:

For ( cos Xi ) is negative, the boundary point (mathrm) is located in the spherical cap whose bottom surface is a circle with diameter (l_>)

For ( cos Xi ) is positive, the boundary point (mathrm) is outside the spherical cap whose bottom surface is a circle with diameter (l_>) .

Here, (l_>) is the length of the line segment (mathrm). In addition, the absolute value of ( cos Xi ) is proportional to the degree of deformation that can be seen from the graph of the cosine function. The authors of Ref. [53] first proposed this kind of quantity for the shadow. It is not difficult to find that . As shown in Fig. 3, any point (mathrm) on the shadow contour corresponds to the unique (cos Xi ) and (Phi /gamma ) . Therefore, considering the symmetry of the shadow, we can obtain the degree of deviation of each point on the shadow contour from the circle through the functional image of (cos Xi ) with respect to (Phi /gamma ) . Now, we can use ( gamma ) and ( Xi ) to represent the sizes and shapes of shadows without confusion.

شاهد الفيديو: مسلسل كوري الثقب الاسود ح 1 قسم 4 (شهر فبراير 2023).