in multi - antenna downlink systems , transmission strategies which require less feedback resources to fully utilize multiuser diversity , but with asymptotic sum capacity comparable to dirty paper coding , are favored .the idea of random beamforming , which satisfies the two aforementioned features has drawn much interest in recent years . in the basic random beamforming strategy suggested in , the transmitter with transmit antennas generates random orthonormal beams and requires each user to feed back the experienced by them for each beam . then the transmitter schedules users for transmission that currently have the best channel for each random beam . despite the considerable literature on this topic ,there are three existing open problems :  1 .  how to address heterogeneous users with diverse large scale channel effects and the impact on scheduling policy ? 2what is the closed form sum rate by exact performance analysis ?3 .  what is the effect of selective feedback , both spatial and spectral , on the randomness of multiuser diversity ?  the first problem is related to a practical downlink system setting with asymmetrically located users having heterogeneous large scale channel effects .this near - far effect was first treated in by observing that the system becomes interference dominated when is large enough . in the large setting , the authors prove that users are asymptotically equiprobable to be scheduled. however , when is finite and not increasing simultaneously with the number of users , the greedy scheduling policy employed in can not maintain fairness among users .also , if a round robin scheduling policy was utilized , fairness can be guaranteed , but no multiuser diversity gain could be achieved for capacity growth . therefore , an alternate scheduling policy is needed to maintain fairness while exploiting multiuser diversity at the same time . in this paper , the cumulative distribution function ( cdf)-based scheduling policy is leveraged and analyzed in the random beamforming framework , wherein the user whose rate for a given beam is high enough but least probable to become higher is selected . under this scheduling policy , each user can be equivalently viewed as competing with other users with the same cdf , thus making the study of individual user rate more relevant and interesting than that of the sum rate . in this paper , we develop the notion of individual sum rate , which is the individual user rate multiplied by the number of users , in order to demonstrate the multiuser diversity gain with user growth for a given user .the second problem addresses exact system analysis , namely deriving closed form expression for the sum rate for arbitrary but finite number of users .note that even with full feedback , wherein each user conveys back the signal - to - interference - plus - noise ratio ( ) for beams , the closed form sum rate has not been derived .this is partially due to the complicated form of and its interplay with multiuser diversity . in this work ,the problem is tackled and solved by a novel probability density function ( pdf ) decomposition which decomposes and interprets the selected user s . in ,the homogeneous setting is considered and in this paper , the technique is extended to the heterogeneous user setting and the closed form individual sum rate is derived .the closed form result under full feedback helps in evaluating the system performance and acts as the building block for exact analysis with selective feedback .the third problem is concerned with standard selective feedback in the spatial dimension , wherein each user feeds back the for the best beam among the beams and the corresponding beam index .this selective feedback is fundamentally different than full feedback in two aspects .the first difference is the two - stage maximization with the first stage maximization carried out by each user for feedback selection and the second stage maximization carried out by the scheduler to perform user selection .since the best beam is selected by each user , the first stage maximization is over correlated .this correlation issue has been addressed in , and the cdf for the selected at the user side is derived . in this paper , we propose an approximation for the cdf and utilize it to derive closed form rate approximation .the other fundamental difference is the number of the values that the scheduler has to maximize over for each beam .this number is fixed and equals the number of users in the full feedback case . however , with selective feedback , it becomes a random quantity .in other words , selective feedback results in a random effect on the multiuser diversity .this effect was first observed in . in this paper, we investigate the randomness of multiuser diversity by extremes over random samples and provide a rigorous argument on the rate scaling .the third problem is further extended to include spectral selectivity by examining a wideband ofdma system , which groups the subcarriers into resource blocks to form the basic scheduling and feedback unit . in order to save feedback resourcewhile not significantly degrading the system performance , additional selective feedback in the spectral dimension is necessary .the effect of random beamforming in a wideband system is examined in by extensive simulations , and further studied from a utility function perspective with the proportional - fair scheduler in . in , analytical results on the asymptotic cluster size is provided . apart from the thresholding - based partial feedback strategy , the best - l selective feedback strategy is appealing and utilized in practical systems such as lte . in this paper, we employ the best - l selective feedback strategy to investigate random beamforming and the effect of spectral dimension selective feedback , which calls for an additional maximization stage at the user side to perform feedback selection . in this feedback strategy , only the best beams from the best resource blocks along with the beam and resource block index are fed back from each user . in this paper, we first derive a closed form rate approximation with exact analysis , i.e. , valid for arbitrary but finite number of users .then , the influence of the additional spectral dimension selective feedback on the type of convergence is investigated with the technique of tail equivalence . moreover, the multiplicative effect of selective feedback and random observations is observed to establish the rate scaling .  to summarize , the main contributions of this paper are threefold : the utilization of cdf - based scheduling policy to address heterogeneous users with random beamforming , the obtained closed form rate results with different selective feedback assumptions , and the asymptotic analysis on the randomness of multiuser diversity incurred by selective feedback .these three contributions analytically examine the raised open problems , and foster further understanding on random beamforming with heterogeneous users and selective feedback .the organization of this paper is as follows .section [ system ] reviews the basic narrowband system model for random beamforming .the analysis for the full feedback case is carried out in section [ full ] , and for the spatial dimension selective feedback in section [ spatial ] .section [ both ] provides the model for the wideband ofdma with random beamforming , and examines the effect of additional spectral dimension selective feedback on rate performance .finally , section [ conclusion ] concludes the paper .we consider a multi - antenna narrowband gaussian downlink channel with single antenna receivers and a transmitter equipped with antennas . a block fading channel model with coherence interval is assumed .the random beamforming strategy employs random orthonormal vectors for , where the s are drawn from an isotropic distribution independently every channel uses . denoting as the transmission symbol at time , the transmitted vector of symbols at time , represented by , is given as let be the received signal at the user , then where is the complex channel vector which is assumed to be known at the receiver , is the additive white noise , and the elements of and are i.i.d .complex gaussian with zero mean and unit variance .note that this channel assumption corresponds to the rayleigh fading assumption for the small scale channel effect . from now on , the time variable will be dropped for notational convenience .the total transmit power is chosen to be , i.e. , =1 ] is the positive part of the argument .note that the distribution does not depend on the selected beam index due to the identically distributed property across beams and is dropped to simplify notation , i.e. , . using a similar procedure to that described in section [ full_rate ] , after receiving feedback, the scheduler performs the transformation for user selection : compared with ( [ full_eq_1 ] ) , it is clear that for the full feedback case .denote as the random variable representing the selected user for beam , then where denotes the set of users conveying feedback for beam . is a set of random size and the probability mass function ( pmf ) can be shown to be given by following the derivation in section [ full_rate ] , let be the selected for beam at the scheduler side , then conditioned on and , the conditional cdf of can be written as . by averaging over the randomness of ,the conditional cdf is expressed as from ( [ full_eq_4 ] ) and ( [ full_eq_6 ] ) , the individual sum rate of user is derived as   with the frchet upper bound and the negative association upper bound for spatial dimension selective feedback ( , db ) . ]  due to the complicated form of , the exact closed form expression for ( [ spatial_eq_6 ] ) is hard to obtain .we now aim to provide an approximate expression for the closed form by examining the property of and utilizing the established result in section [ full_rate ] .recall that is the maximization over correlated random variables , thus alternative approximation for would lead to rate approximation .one simple approach is to use the frchet upper bound for the s . since the s are identically distributed across , the frchet upper bound yields .this upper bound is very loose empirically for .one suitable approach is inspired by the conjectured negative associated upper bound proposed in to deal with the minimum mean square error ( mmse ) receiver .our empirical evidence shows that even with single antenna receiver , the s are negative associated , thus the upper bound produced by the negative association property can be utilized to approximate , namely fig .[ fig_1 ] illustrates the bounds and the empirical cdf for , db .it can be seen that the proposed upper bound in ( [ spatial_eq_7 ] ) approximates the exact one in ( [ spatial_eq_1 ] ) well , especially when the is large . by using the cdf approximation ,the individual sum rate can be approximated by a closed form expression presented in the following corollary .[ corollary_1 ] ( _ closed form approximation of individual sum rate _ )   the proof is given in appendix [ appenb ] .  in order to demonstrate the rate approximation in corollary [ corollary_1 ], we conduct a numerical study in fig .[ fig_2 ] for different and with respect to the number of users .the exact in ( [ spatial_eq_6 ] ) can be calculated by numerical integration .it is observed that ( [ spatial_eq_8 ] ) approximates the exact rate very well , which makes the rate approximation valuable due to its efficient computational form .   and with respect to the number of users ( , db , db , db ) . ]the difficulty of dealing with rate scaling with selective feedback is two - fold .firstly , due to selective feedback of the best beam , the number of to maximize over at the scheduler side for each beam is a random quantity .this random effect is reflected in the random set in section [ spatial_rate ] .secondly , the normalizing constants for establishing the type of convergence have to be obtained for a quantity other than the number of users in the full feedback case . in , the first issue was tackled by the delta method . in this paper , we solve the first issue by referring to the extremes over random samples , and rigorously solve the second one by using the normalizing constants theorem .the proof is provided in appendix [ appenb ] .  to examine the random effect on multiuser diversity ,denote the sequence of random variables as the number of fed back for beam with users .it is easy to see that are binomial distributed with probability of success .thus by the strong law of large numbers , as grows , the number of fed back for each beam becomes .the following theorem is called upon to deal with this random effect .[ theorem_3 ] ( _ extremes with random sample size ) _ let , as , in probability , where is a positive random variable .assume that there are sequences such that converges weakly to a nondegenerate distribution function .then , as ,   therefore , if we denote as the extreme order statistics of the received for each beam of a given user , then from theorem [ theorem_3 ] , its cdf can be efficiently approximated by .combining this with the normalizing constants theorem in appendix [ appenb ] yields the following corollary .[ corollary_2 ] ( _ individual scaling laws under spatial dimension selective feedback _ )   the proof is given in appendix [ appenb ] ._ remark : _ the scaling for the exact rate and approximated rate differs in the factor .the rate scaling for does not have this factor because intuitively the exponent in the approximated cdf counteracts the reduction in the number of values for maximization , i.e. , , due to selective feedback .we call this effect as the _ multiplicative effect_. the detailed proof can be found in appendix [ appenb ] . to draw further insights, we can think of the exponent of as the _ virtual _ users . in the full feedback case ,the exponent equals . in the selective feedback case with the approximated cdf , the exponent asymptotically equals by the aforementioned multiplicative effect .the notion of virtual users and the multiplicative effect will be investigated further with both spatial and spectral dimension selective feedback in section [ both_scaling ] .  , resource blocks , beams , the spectral dimension selective feedback ) . ]in this section , random beamforming is embedded in a wideband ofdma system .the system model is presented in section [ both_model ] , the exact analysis and the asymptotic analysis are examined in section [ both_rate ] and section [ both_scaling ] respectively .   the system model described in section [ system ]is extended to an ofdma system with resource blocks .each resource block is regarded as the basic scheduling and feedback unit .the random beamforming strategy generates orthonormal beams for each resource block .denote as the transmission symbol at resource block , then the received signal for user at resource block can be expressed as where is the frequency domain channel transfer function of user at resource block with i.i.d . elements . to facilitate analysis , is assumed to be i.i.d . across resource blocks for a given user .this corresponds to the widely used block fading approximation in the frequency domain due to its simplicity and capability to provide a good approximation to actual physical channels .the transmit power for a resource block is assumed to be . from ( [ both_eq_1 ] ) ,the of user at resource block for beam is , and is denoted by for notational simplicity . for a given user , the s are i.i.d . across resource blocks for a given beam , and for a given resource block , the s are identically distributed and correlated across beams .the cdf of is given by , where the index and can be dropped due to the identically distributed property .with the extra degrees of freedom in the spectral dimension , additional selective feedback at each user side can be made possible by the following two - stage feedback selection .the first stage selection is in the spatial dimension , where each user selects the best beam with the largest for each of the resource block .this process is similar to the narrowband feedback selection discussed in section [ spatial_rate ] .let be the outcome of the first stage selection , thus from ( [ spatial_eq_1 ] ) , its cdf can be written as +^me^{-\frac{2mx}{\rho_kd_{\imath}(x)}}}{a_{\imath}(x)}\right)u(x)$ ] , where again the resource block index and the beam index can be dropped due to the identically distributed property across resource blocks and beams .the second stage selection occurs in the spectral dimension , where each user feeds back the values of the best resource blocks among the total resource blocks .let denote the outcome of the second stage selection of user at resource block for beam .thus this random variable represents the selected at the user side , whose cdf is of interest for further analysis .it is easy to see that for the case of full feedback in the spectral dimension , i.e. , , . for the best-1 feedback case , i.e. , , due to the independent property of across resource blocks .for the general best - l feedback case , utilizing the results in , the cdf can be shown as where .the two - stage feedback selection is demonstrated in fig .[ fig_3 ] with nine users denoted by different colors , five resource blocks , and four beams . in the illustrated example, we use best-2 spectral dimension feedback , i.e. , .  after receiving feedback, the scheduler performs the cdf - based scheduling by first conducting the transformation on the received , denote as the random variable representing the selected user at resource block for beam , then where denotes the set of users conveying feedback for beam at resource block .following the derivation in section [ spatial_rate ] , let be the selected for beam at resource block at the scheduler side .then averaging over the randomness of , the conditional cdf conditioned on can be written as for further derivation , is manipulated into the following form by the power series expansion : where following the same procedure as in section [ spatial_rate ] , the individual sum rate for user can be derived as \notag\\ & = m\sum_{\tau_1=1}^k{k\choose \tau_1}\left(\frac{1}{m}\right)^{\tau_1}\left(1-\frac{1}{m}\right)^{k-\tau_1}\sum_{\tau_2=1}^{\tau_1}{\tau_1\choose \tau_2}\left(\frac{l}{n}\right)^{\tau_2}\left(1-\frac{l}{n}\right)^{\tau_1-\tau_2}\notag\\ \label{both_eq_8}&\quad\times \sum_{\ell=0}^{\tau_2(l-1)}\xi_2(n , l,\tau_2,\ell)\int_0^{\infty}\log_2(1+x)d(f_{y_k}(x))^{n\tau_2-\ell}.\end{aligned}\ ] ] in order to obtain the closed form rate approximation for , the cdf approximation proposed in ( [ spatial_eq_7 ] ) by the negative association property is utilized to approximate .the closed form result is presented in the following corollary .[ corollary_3 ] ( _ closed form approximation of individual sum rate _ )   to understand the impact of spectral dimension selective feedback , we conduct a numerical study assuming , .[ fig_4 ] plots the exact and approximated rate for different under db with respect to the number of users .it can be seen that when the number of users is small , there is a certain rate gap between selective feedback and full feedback .however , the gap becomes negligible when the number of users increases .[ fig_5 ] , the performance is observed for different for . from the two figures , we can see that the proposed rate approximation tracks the exact performance very well .with respect to the number of users ( , , db , ) . ]   with respect to different ( , , , ) . ]we now examine the rate scaling with selective feedback in both spatial and spectral dimension . in section [ spatial_scaling ] with spatial dimension selective feedback ,the cdf of interest is and the number of to maximize over at the scheduler side for each beam approaches . with additional spectral dimension feedback ,the cdf of is of primary interest . to get a handle on the randomness of multiuser diversity for this case , an approach similar to that in section [ spatial_scaling ] can be utilizedlet the sequence of random variables be the number of values fed back for beam at resource block with users .it is easy to see that are binomial distributed with probability of success .therefore , by the strong law of large numbers , as grows , the number of values fed back for each beam at each resource block becomes .moreover , the convergence property of the sequence can be shown by invoking the central limit theorem : where indicates convergence in distribution . by applying theorem [ theorem_3 ] ,the extreme order statistics of the received for each beam at each resource block for a given user can be efficiently approximated by .   and its tail equivalence for different spectral dimension selective feedback ( , , db , ) . ]now the remaining problem is to examine the type of convergence of .recall the formulation of as : .it is known that converges weakly to the type gumbel distribution . due to the complicated form of ,it is tedious to directly check the conditions for proving the type of convergence . in order to investigate the tail behavior of which dominates the type of convergence , the following tail equivalence theorem is called upon .  [ theorem_4 ]( _ the tail equivalence theorem ) _ and are distribution functions such that if there exist normalizing constants , such that , where is non - degenerate , then .from theorem [ theorem_4 ] one can infer that if two distribution functions are tail equivalent , then they belong to the domain of attraction of the same type .employing theorem [ theorem_4 ] , a tail equivalent formulation can be obtained for expressed in the following corollary .[ corollary_4 ] ( _ tail equivalent cdf _ ) is tail equivalent to .the proof is given in appendix [ appenc ] .  fig .[ fig_6 ] compares the exact cdf and the corresponding tail equivalence for different selective feedback under , , and db . the tail equivalent cdf is observed to track the exact one even when is small , which supports and lends confidence in the power of the tail equivalence theorem .therefore , the tail equivalence is used to study the type of convergence , which is expressed in the following lemma .[ lemma_2 ] ( _ type of convergence of selective feedback _ ) given the statistical property of in ( [ spatial_eq_1 ] ) , belongs to the domain of attraction of type gumbel distribution .the proof is given in appendix [ appenc ] .having obtained the type of convergence for , the rate scaling result can be derived by referring to the normalizing constants theorem in appendix [ appenb ] .the individual rate scaling is provided below .[ theorem_5 ] ( _ individual scaling laws under spatial and spectral dimension selective feedback _ )   the proof is given in appendix [ appenc ] .  _remark : _ for the exact rate , the ultimate equivalent cdf of interest is , thus the exponent due to multiplicative effect can be seen as the virtual users for scheduling competition .this exponent is for the general best - l spectral dimension feedback . for the full feedback ,since equals for and otherwise , the cdf becomes . for the best-1 feedback , since is for and otherwise , the cdf becomes .intuitively , the best-1 feedback is asymptotically optimal due to the same number of virtual users .in other words , even though additional maximization reduces the average number of variables for feedback , it counteracts this reduction by increasing the exponent of the cdf .the number of virtual users is the limiting factor that dominates rate scaling . for the approximated rate ,since the approximated cdf compensates for the spatial dimension selection by increasing the exponent , the rate scaling differs by a factor of .in this paper , an analytical approach is used to investigate the problem of random beamforming with heterogeneous users and selective feedback .the heterogenous user scenario corresponds to the practical scenario of potentially different large scale channel effects for different users .we leverage the cdf - based scheduling policy to decouple each user s rate and thus theoretically examine the individual user rate .we develop the notion of individual sum rate to analyze the rate scaling for each individual user .we focus our analysis in this work on theoretically understanding the effect of selective feedback in both spatial and spectral dimensions . on the exact analysis part ,extensive numerical results show that our approximate expression for the rate under selective feedback is effective and provides an efficient expression for computing the exact rate . on the asymptotic analysis part ,we develop the notion of virtual users and the multiplicative effect to explain the impact of selective feedback on rate scaling .we further discover that the limiting factor for the rate scaling is the exponent for the ultimate cdf of the selected at the scheduler side .the extension of this work from single antenna users to multiple antenna users and more generally the investigation of a multicell mimo setup with advanced receiver design will be the subject of our future work ._ proof of theorem [ theorem_1 ] : _ with the help of lemma [ lemma_1 ] , can be computed as where ( a ) follows from applying lemma [ lemma_1 ] ; ( b ) follows from integration by parts . the closed form result for in theorem [ theorem_1 ]can be computed in a recursive manner and is presented as follows where is the exponential integral function of the first order ._ proof of corollary [ corollary_2 ] : _ it is shown in that belongs to the domain of attraction of type gumbel distribution . thus if the number of to maximize over for each beam is fixed and equals the number of users , then the following equation holds : , where is the type gumbel distribution , and represent the normalizing constants for user . from theorem [ theorem_3 ] ,the number of to maximize over for each beam approaches .let and denote the normalizing constants for user under the selective feedback case .then the following equation holds : . in order to obtain and , the following theorem is called upon .  [ theorem_6 ]( _ the normalizing constants theorem ) _ let be a sequence of distribution functions .let , , , and be sequences of real numbers such that , as , for all continuity points of the limits , where and are nondegenerate distribution functions . then , as ,the limits : , are finite , and .  the spatial dimension selective feedback case possesses the following situation in theorem [ theorem_6 ] : , , , , , , and .the sequence of has been derived in as : .a suitable choice of for type is , where is the growth function for user defined by .thus a suitable sequence is for all . solving yields , . therefore , by referring to theorem [ theorem_6 ] , the normalizing constants can be derived to be : , and for all . then by employing the corollary a.1 . in , the individual rate for user , namely scales as .regarding the approximated rate , since the approximated cdf by negative association is and the number of to maximize over approaches , we have .thus the normalizing constants , and , which enables the approximated rate to scale as .    _ proof of lemma [ lemma_2 ] : _ with statistics in ( [ spatial_eq_1 ] ) belongs to the domain of attraction of type .it can be shown that for any distribution function which converges weakly to the limiting distribution , then its exponent form has the same type of convergence , belongs to the domain of attraction of type .then by theorem [ theorem_4 ] , belongs to the domain of attraction of type ._ proof of theorem [ theorem_5 ] : _ a procedure similar to that used in proving corollary [ corollary_2 ] can be used here . since the number of to maximize over for each beam at each resource block approaches , and belongs to the domain of attraction of type , the following equation holds : . by referring to the tail equivalence theorem , the equivalent equation is : .applying theorem [ theorem_6 ] yields the normalizing constants : , and for all .therefore , scales as .    d. j. love , r. w. heath , v. k. n. lau , d. gesbert , b. d. rao , and m. andrews , `` an overview of limited feedback in wireless communication systems , '' _ ieee j. sel .areas commun ._ , vol . 26 , no . 8 ,13411365 , oct .2008 .r. knopp and p. a. humblet , `` information capacity and power control in single - cell multiuser communications , '' in _ proc .ieee international conference on communications ( icc ) _ , jun .1995 , pp . 331335 .s. vishwanath , n. jindal , and a. goldsmith , `` duality , achievable rates , and sum - rate capacity of gaussian mimo broadcast channels , '' _ ieee trans .inf . theory _ ,49 , no .10 , pp . 26582668 , oct .2003 .h. weingarten , y. steinberg , and s. shamai , `` the capacity region of the gaussian multiple - input multiple - output broadcast channel , '' _ ieee trans .inf . theory _ ,52 , no . 9 , pp . 39363964 , sept .2006 .h. wang and t. kirubarajan , `` random unitary beamforming with partial feedback for miso downlink transmission using multiuser diversity , '' _ ieee trans . signal process ._ , vol .55 , no . 7 ,33803389 , jul . 2007 .k. park , y. ko , and m. alouini , `` accurate approximations and asymptotic results for the sum - rate of random beamforming for multi - antenna gaussian broadcast channels , '' in _ proc .ieee vehicular technology conference ( vtc ) _ , apr .2009 , pp ..  s. h. moon , s. r. lee , and i. lee , `` sum - rate capacity of random beamforming for multi - antenna broadcast channels with other cell interference , '' _ ieee trans .wireless commun ._ , vol . 10 , no . 8 , pp . 24402444 , aug .d. park , h. seo , h. kwon , and b. g. lee , `` wireless packet scheduling based on the cumulative distribution function of user transmission rates , '' _ ieee trans ._ , vol . 53 , no . 11 , pp . 19191929 , nov .2005 .p. svedman , s. k. wilson , l. j. cimini jr , and b. ottersten , `` a simplified opportunistic feedback and scheduling scheme for ofdm , '' in _ proc .ieee vehicular technology conference ( vtc ) _ , vol . 4 , may 2004 , pp .18781882 .yichao huang ( s10m12 ) received the b.eng .degree in information engineering with highest honors from the southeast university , nanjing , china , in 2008 , and the m.s . and ph.d .degrees in electrical engineering from the university of california , san diego , la jolla , in 2010 and 2012 , respectively .he then join qualcomm , corporate r&d , san diego , ca .he interned with qualcomm , corporate r&d , san diego , ca , during summers 2011 and 2012 .he was with california institute for telecommunications and information technology ( calit2 ) , san diego , ca , during summer 2010 .he was a visiting student at the princeton university , princeton , nj , during spring 2012 .huang received the microsoft young fellow award in 2007 from microsoft research asia .he received the ece department fellowship from the university of california , san diego in 2008 , and was a finalist of qualcomm innovation fellowship in 2010 .his research interests include communication theory , optimization theory , wireless networks , and signal processing for communication systems .bhaskar d. rao ( s80m83sm91f00 ) received the b.tech .degree in electronics and electrical communication engineering from the indian institute of technology , kharagpur , india , in 1979 , and the m.sc . and ph.d .degrees from the university of southern california , los angeles , in 1981 and 1983 , respectively .since 1983 , he has been with the university of california at san diego , la jolla , where he is currently a professor with the electrical and computer engineering department .he is the holder of the ericsson endowed chair in wireless access networks and was the director of the center for wireless communications ( 20082011 ) .his research interests include digital signal processing , estimation theory , and optimization theory , with applications to digital communications , speech signal processing , and human computer interactions .rao s research group has received several paper awards .his paper received the best paper award at the 2000 speech coding workshop and his students have received student paper awards at both the 2005 and 2006 international conference on acoustics , speech , and signal processing , as well as the best student paper award at nips 2006 .a paper he coauthored with b. song and r. cruz received the 2008 stephen o. rice prize paper award in the field of communications systems .he was elected to the fellow grade in 2000 for his contributions in high resolution spectral estimation .he has been a member of the statistical signal and array processing technical committee , the signal processing theory and methods technical committee , and the communications technical committee of the ieee signal processing society .he has also served on the editorial board of the eurasip signal processing journal .

Abstract: this paper investigates three open problems in random beamforming based communication systems : the scheduling policy with heterogeneous users , the closed form sum rate , and the randomness of multiuser diversity with selective feedback . by employing the cumulative distribution function based scheduling policy , we guarantee fairness among users as well as obtain multiuser diversity gain in the heterogeneous scenario . under this scheduling framework , the individual sum rate , namely the average rate for a given user multiplied by the number of users , is of interest and analyzed under different feedback schemes . firstly , under the full feedback scheme , we derive the closed form individual sum rate by employing a decomposition of the probability density function of the selected user s signal - to - interference - plus - noise ratio . this technique is employed to further obtain a closed form rate approximation with selective feedback in the spatial dimension . the analysis is also extended to random beamforming in a wideband ofdma system with additional selective feedback in the spectral dimension wherein only the best beams for the best - l resource blocks are fed back . we utilize extreme value theory to examine the randomness of multiuser diversity incurred by selective feedback . finally , by leveraging the tail equivalence method , the multiplicative effect of selective feedback and random observations is observed to establish the individual rate scaling .   ieee transactions on wireless communications , to appear  random beamforming , multi - antenna downlink channels , heterogeneous users , selective feedback , individual sum rate , individual scaling laws , multiuser diversity


the schwarzschild solution has proved to play a fundamental importance in conceptual discussions of general relativity , and beyond , for instance , regarding event horizons , spacetime singularities and aspects of quantum field theory in curved spacetimes .it has also been important providing the first insights regarding the phenomenon of gravitational collapse and inspired the construction of theoretical models of relativistic stars . before the mid-1960s , the object now known as a black hole , was referred to as a collapsed star or as a frozen star , and it was only in 1965 that marked an era of intensive research into black hole physics .relatively to the issue of experimental tests of the schwarzschild solution , the exterior geometry has been extremely successful in explaining , for instance , the precession of mercury s perihelion , and the phenomenon of the bending of light , where the exterior schwarzschild gravitational field acts as a gravitational lens .  despite of its important role ,one still encounters , in the literature , the existence of misconceptions and a certain ambiguity inherent in the schwarzschild solution .for instance , a problematic aspect is the presence of an event horizon , which in the schwarzschild black hole solution acts as a one - way membrane , permitting future - directed null or timelike curves to cross only from the exterior to the interior region .it acts as a boundary of all events which , in principle , may be observed by an exterior observer .it is believed that the gravitational collapse of a compact body results in a singularity hidden beyond an event horizon . if the singularity were visible to the exterior region , one would have a _ naked singularity _ , which would open the realm for wild speculation .this led to penrose s _ cosmic censorship conjecture _ , which stipulates that all physically reasonable spacetimes are globally hyperbolic , forbidding the existence of naked singularities , and only allowing singularities to be hidden behind event horizons .the cosmic censorship conjecture has been an active area of research and the source of considerable controversy . for the interior black hole solution ,a remarkable change occurs in the nature of spacetime , namely , the external spatial radial and temporal coordinates exchange their character to temporal and spatial coordinates , respectively . thus , the interior solution represents a non - static spacetime , as the metric coefficients are now time - dependent .this also implies that a singularity occurs at a spacelike hypersurface , .thus , no observer , interior or exterior to the schwarzschild radius , will be able to observe the formation , or for that matter , the physical effects of the singularity .these aspects show the existence of inconsistencies and a certain ambiguity inherent in the schwarzschild solution .  still relatively to the issue of the black hole event horizon, a widespread misconception in the literature is that a test particle approaches the schwarzschild radius at the speed of light for all observers , and not as a limiting process for a static observer located at the event horizon given by the null hypersurface , where is the black hole mass .we shall use geometrized units , i.e. , , for notational convenience , throughout this paper .if one accepts that a particle has the speed of light with respect to a static observer , at , then using the local special relativity velocity composition law , the observer concludes that the particle has the speed of light with respect to all observers , which is another way of saying that in the frame of a photon all particles have speed .of course , the frame of the photon is not a physical frame .indeed , it should be emphasized that an observer can not remain at rest at , as it implies an infinite acceleration to do so .another erroneous extrapolation of at , is that timelike particles attain velocities superior to the speed of light in the black hole region , and thus justifying the existence of tachyons , or for that matter of particles with velocities superior to the speed of light in the framework of general relativity .all things considered , it should be stressed once again that one can only use static observers in the spacetime region characterized by . andit was shown that the test particle does indeed cross the event horizon with a velocity less than the speed of light with respect to an appropriate physical observer .an exact general expression was further deduced , in terms of the constants of motion of both a test particle and an observer moving along radially infalling geodesics , given by where and are the constants of motion for the test particle and the observer , respectively .this expression shows that the value of the velocity at is indeed less than , unless or are zero or infinite ( see ref . for details , and ref . for further discussions ) .relatively to spacetime singularities , a key aspect is whether they are a disaster for the theory , as they imply the breakdown of predictability .various definitions of singularities can be encountered in the literature , ranging from specific invariants , constructed from the curvature tensor and its covariant derivatives , becoming infinite , to the notion of geodesic incompleteness .accordingly , one encounters several attitudes to spacetime singularities : firstly , singularities are mere artifacts of unrealistic and idealized models where infinities occur .secondly , general relativity entails singularities , according to the penrose - hawking theorems in the context of geodesic incompleteness , and fails to accurately describe nature .thirdly , one may have a more optimistic viewpoint , as expressed by misner , who views the existence of singularities , not as proof of our ignorance , but as a source from which we can derive much valuable understanding of cosmology" .much of the scepticism related to the concepts of event horizons and spacetime singularities , outlined above , and others such as the information paradox , has inspired new and fascinating ideas , namely by replacing the interior solution , and thus , doing away with the problems related to these issues . in this context , it is interesting to note the emergence of a new picture for an alternative final state of gravitational collapse , where an interior compact object is matched to an exterior schwarzschild vacuum spacetime , at or near where the event horizon is expected to form .these alternative models do not possess a singularity at the origin and have no event horizon , as its rigid surface is located at a radius slightly greater than the schwarzschild radius . in particular , the gravastar ( _ grav_itational _ va_cuum _ star _ ) picture , proposed by mazur and mottola , has an effective phase transition at / near where the event horizon is expected to form , and the interior is replaced by a de sitter condensate .it has also been argued that there is no way of distinguishing a schwarzschild black hole from a gravastar from observational data . however , a realistic model for the gravitational collapse leading to compact interior objects is still lacking .we also emphasize that , although interesting in themselves , the solutions that do away with the interior singularity and the event horizon sweep the inherent conceptual difficulties of black holes under the rug , and that the interior structure of realistic black holes has not been satisfactorily determined , being still open to considerable debate .  in this work , we shall review and analyze the interior schwarzschild solution in some detail . in the literature ,the interior geometry is considered as a continuity of the exterior geometry .therefore , for instance , infalling test particles are considered ending up at a central singularity located at . however , the latter singularity is a spacelike hypersurface , and the test particles are not directed towards a privileged point , but directed along a temporal direction , in order to not violate causality .thus , if one looks at the interior geometry as a continuation of the exterior static solution , one comes across some extremely interesting conceptual difficulties , that question the very concept of a black hole . in this workwe shall address some of these difficulties .we shall start by obtaining the metric for the interior region without inferring directly to the traditional schwarzschild solution . with this geometry at hand, one may analyze this interior region , without the prejudices inherited from the exterior region . however , we emphasize that we shall compare both regions where it is convenient to do so .we find new interesting features and peculiarities not encountered in the literature , and show that this scenario can be interpreted as a cosmological solution .this paper is outlined in the following manner : firstly , in section [ sec : interior ] , we deduce the interior metric by considering time - dependent metric fields , paying close attention to several cosmological interpretations of this solution .we also address some of the difficulties associated to spacetimes singularities , and argue that it is perhaps possible for an observer to know if he is inside or outside the schwarzschild radius , by examining an invariant of a covariant derivative of the curvature tensor .secondly , in sections [ sec : conserved ] and [ sec : geodesics ] , we deduce the conserved quantities of null and timelike geodesics in some detail , discussing several particular cases .thirdly , in sections [ sec : ek - coordinates ] and [ sec : kruskal - coordinates ] , we analyze the eddington - finkelstein and kruskal coordinates directly from the interior solution .finally , we conclude in section [ sec : conclusion ] .we shall be interested in the spacetime metric given by where and are the spacetime coordinates .assume that , so that the line element with and describes a with an area given by .in particular , we shall consider the specific case of .  to set the nomenclature , note that the mathematical analysis and the physical interpretation will be simplified using a set of orthonormal basis vectors .these may be interpreted as the proper reference frame of a set of observers who remain at rest in the coordinate system , with fixed .denote the basis vectors in the coordinate system as , , , and .then , using the following transformation , , with \,,\ ] ] one finds in this basis the metric components assume their minkowskian form , .  the einstein field equation , in an orthonormal reference frame , is given by where is the stress - energy tensor and is the einstein tensor , given by . is the ricci tensor , which is defined as a contraction of the riemann ( or curvature ) tensor , , and is the scalar curvature defined as a contraction of the ricci tensor , .the einstein tensor , given in the orthonormal reference frame , , yields for the metric ( [ metric ] ) the following non - zero components \ , , \label{einsteintheta}\\ g_{\hat{\phi}\hat{\phi}}&=&g_{\hat{\theta}\hat{\theta}}\ , , \label{einsteinphi}\end{aligned}\ ] ] where a prime denotes a derivative with respect to the coordinate , and the overdot a derivative with respect to the temporal coordinate , .we shall consider a vacuum solution , i.e. . from the addition of eqs .( [ einsteintt ] ) and ( [ einsteinzz ] ) , we verify so that the solution , with is obtained .it is possible to absorb the function , defining a new spatial coordinate , so that without a significant loss of generality one may set .one may conclude from this analysis that .  from eq .( [ einsteinzt ] ) , we verify , so that , implying .note that is related to the proper time , so that one may impose that .now , substituting the relationship into eq .( [ einsteintt ] ) , one finally deduces that where is a constant of integration with time dimension . from the condition , andconsequently , this solution is only valid for .  defining the constant of integration as the metric ( [ metric ] ) finally takes the form the constant may be determined from a direct confrontation with the exterior schwarzschild solution , and is given by , where is the black hole mass .it may take the physical significance of a characteristic time for the existence of universes in the interior schwarzschild solution , as may be inferred from the cosmological interpretation of the interior metric , given by eq .[ metric ] , which we consider in the next subsection .this interior solution illustrates a particularly strange , yet physically meaningful picture of the universe within the event horizon .thus , we shall consider some interesting astrophysical and cosmological interpretations of this solution . a quick glance at the metric ( [ intmetric ] ) is enough to convince one that this also corresponds to an anisotropic and homogeneous cosmological solution .in fact , considering the kantowski - sachs solution given by where and are the scale factors of the geometry , one verifies that both metrics are identical , by taking into account the following transformation   an alternative approach would be to consider a time - dependent parameter  , so that one could generalize metric ( [ intmetric ] ) to ^{-1}\ , dt^2+\left[\frac{2\xi(t)}{t}-1 \right]dz^2   \nonumber  \\ & & + t^2 \,(d\theta ^2 + \sin ^2{\theta } \ , d\phi ^2 )  \ , .\label{gen - intmetric}\end{aligned}\ ] ] the einstein tensor given in an orthonormal reference frame has the following non - zero components where the overdot denotes a derivative with respect to the time coordinate , as before . note that this solution implies , where and are the energy density and the pressure along the , much in the spirit of refs .note that the geometry ( [ gen - intmetric ] ) where is time - dependent is not a solution of the vacuum einstein equations .in addition , even though , they are time - dependent , i.e. , the corresponding `` cosmological constant '' is not constant at all .the above - mentioned case provides some very interesting cosmological solutions , in rather different contexts , however , they shall be presented elsewhere .several cosmological scenarios have also been proposed , in which a universe emerges from the interior of a black hole ( see , for instance , ref . and references therein ) . in the present work, we shall only consider several interesting interpretations of universes within the schwarzschild radius , relatively to the metric ( [ intmetric ] ) .consider the interior solution as measured by an observer at rest relatively to the space coordinates , i.e. , .in this case , from the metric ( [ intmetric ] ) , we have for the positive sign , we have the solution + \frac{\pi\xi}{2}\ , ,   \label{prop - time}\ ] ] where the constant of integration has been chosen to provide for . note that as , we have , so that as coordinate time increases , the proper time as measured by observers at rest also increase .the evolution of this universe may be further explored by considering the negative sign of eq .( [ int - observer ] ) , which yields the following solution + \frac{3\pi \xi}{2}\ , .\label{prop - time2}\ ] ] the constant of integration has been chosen to provide for .for this case the coordinate time decreases from to , however , proper time increases from to .this behavior is represented in fig .[ fig : propertime ] .coordinate system .the evolution of a universe begins at , and proper time .as coordinate time flows from to , proper time runs from to .this corresponds to the positive sign solution of eq .( [ prop - time ] ) .now , allowing for the time coordinate to flow backwards from to , proper time as measured by observers at rest , inexorably runs forward from to .this corresponds to the negative sign solution of eq .( [ prop - time2 ] ) .see the text for details.,width=307 ]  taking into account the metric ( [ intmetric ] ) , we verify that it possesses rotational invariance , as the spatial surfaces corresponding to , represent a with an area given by .the proper distance between two simultaneous events along a determined spatial direction , for instance and , is given by note that a singularity occurs for , as can also be verified from the curvature tensor .the proper distance between two particles at rest separated by a constant , decreases along the as coordinate time flows from to , and increases as coordinate time flows backwards from to .despite the fact that one may not talk about an asymptotic limit , for the interior solution , it is interesting to note that the spacetime assumes an instantaneous minkowski form , for , although the curvature does not become zero .the proper distance between two simultaneous events along a spatial trajectory with and , is given by which increases as varies from to , and decreases when the temporal coordinate runs backwards from to .therefore , one may conclude by assuming the evolution of a universe beginning at , where a singularity occurs along the , however , with no extension along the angular direction , .as coordinate time flows from to , observers at rest move together , i.e. , their proper distance decreases to zero , and move apart along the angular coordinate attaining a maximum at .now , allowing for the coordinate to flow backwards from to , proper time as measured by observers , at rest relatively to the coordinate system , inexorably runs forward from to . for this case , observers move apart along the and collapse along the angular coordinate .  in this example , the difference of choosing an interior observer , without the prejudices inherited from the exterior geometry , is striking .while for the exterior observer , infalling particles tend to a central singularity , from the interior point of view , the proper distance along the increases , showing the existence of a cigar - like singularity .we emphasize that the latter occurs along a spacelike hypersurface .another difference worth mentioning is that the exterior observer considers a spherically symmetric geometry , while the interior observer may consider the geometry plane , as points for different are parallel to one another ( see ref . for details regarding this issue ) .the schwarzschild solution has played a fundamental role in conceptual discussions of general relativity , in particular , regarding spacetime singularities , as mentioned in the introduction .a key aspect of singularities in general relativity is whether they are a disaster for the theory , as it implies the breakdown of predictability .attitudes in the literature range from : singularities are mere artifacts of unrealistic and idealized models ; general relativity entails singularities , but fails to accurately describe nature ; and one may view the existence of singularities `` not as proof of our ignorance , but as a source from which we can derive much valuable understanding of cosmology '' , quoting misner in the latter attitude .  a way of detecting singularities is to find where the energy density or the spacetime curvature become infinite and the usual description of the spacetime breaks down .however , to be sure that there is an essential singularity which can not be transformed away by a coordinate transformation , invariants are constructed from the curvature tensor , such as , , , and from certain covariant derivatives of the curvature tensor .for instance , in the schwarzschild spacetime there is an essential curvature singularity at in the sense that along any non - spacelike trajectory falling into the singularity , as , the so - called kretschman scalar tends to infinity , i.e. , , as shall be shown below . in this case , however , all future directed non - spacelike geodesics which enter the event horizon at must fall into this curvature singularity within a finite value of the affine parameter .so , all such curves are future geodesically incomplete . for the black hole region , given by the metric ( [ intmetric ] ), the scalar kretschmann polynomial , , is given by showing that a curvature singularity occurs at .it is remarkable that a change of sign occurs in the following scalar , as an observer traverses the event horizon note that the invariant is zero on the horizon .it is perhaps possible that this invariant is devoid of a fundamental significance .however , it is generally known that using the curvature tensor and some of its covariant derivatives , the analysis gives a complete description of the geometry , and are directly measurable .since these quantities are coordinate invariant , the problems associated with a specific choice of the coordinate system vanish .this argument may be used in favor of separating the interior from the exterior region .the tidal acceleration felt by an observer at rest is given by where is the observer s four velocity and is the separation between two arbitrary parts of his body .note that is purely spatial in the observer s reference frame , as , implying . is the riemann tensor , given in the orthonormal reference frame , and has the following non - zero components taking into account the antisymmetric nature of in its first two indices , we verify that is purely spatial with the components finally , using the components of the riemann tensor , the tidal acceleration has the following components note a stretching along the , and a contraction along the orthogonal directions .these stretchings and contractions are now time - dependent , contrary to their counterparts in the exterior region , and as , the tidal forces diverge .consider the euler - lagrange equations where the overdot here represents a derivative with respect to the affine parameter defined along the geodesic , which has the physical interpretation of a proper time for timelike geodesics .consider the following lagrangian if the metric tensor does not depend on a determined coordinate , , one obtains an extremely important result .for this case , eq .( [ euler - lagrange ] ) reduces to this implies that the quantity given by is constant along any geodesic . using the lagrangian nomenclature , one denotes a cyclic coordinate , and the respective conjugate momentum .the existence of cyclic coordinates allows one to obtain integrals of the geodesic equation , and provides certain quantities that are conserved along the movement of the particle .  applying the above analysis to the line element ( [ intmetric ] ) ,one verifies that the metric tensor is independent of the coordinates and , so that the conserved quantities are given by  may be interpreted as the angular momentum per unit mass , and possesses the dimensions of a velocity .as may take any real value , we shall consider it as a mere conserved quantity , without any physical significance .the line element ( [ intmetric ] ) may be rewritten in terms of the constants defined above , for the particular case of , in the following manner where is defined for null geodesics , and for timelike geodesics .  for timelike geodesics , , the conserved quantities and may also be determined from the initial conditions . for this purposeit will prove useful to provide an intrinsic definition of velocity , which we shall include next for self - completeness .consider the four - velocity , , tangent to the worldline of an observer , and a four - dimensional spacetime , , orthogonal to .define the operator which has the property of projecting any four - vector on the tangent space of the hypersurface , , so that .thus , one may express the metric tensor in the following form the quantity is the projection of the displacement of a particle , , along the velocity of the observer , so that the particle has a displacement of , along .thus , the velocity may then be defined as   now , consider that the observer is at rest in the coordinates , so that his / her four - velocity is given by , with .thus , we have and so that eq .( [ velocity ] ) may be finally written as this result is identical to the one obtained by landau and lifschitz .now , using the metric ( [ intmetric ] ) and considering , eq .( [ velocity2 ] ) takes the form and finally using eq .( [ line1 ] ) , we have   considering the particular case of , i.e. , fo a test particle that moves along the eq .( [ gen - geod ] ) takes the form the qualitative behavior for the positive values of , in the parameter space of and , is represented in fig .[ fig : pz - along - z ] .note that may take arbitrarily large values as or as .   in the parameter space and , where we have considered .see the text for details.,width=288 ]  for a point particle with an initial velocity and initial time , then is given by . note that if the test particle is at rest , , at an instant , then it will always remain at rest as .if the particle came in from the exterior region , it possesses a conserved quantity along its geodesic .despite the fact that after the crossing of the event horizon its character changes into a constant with the dimensions of a velocity , its numerical value is conserved , i.e. , . the constant may assume different positive values depending on its initial conditions .however , as reflected by eq .( [ pz ] ) , may assume negative values as well , so that one may conclude that geodesic particles moving along a decreasing coordinate , and increasing coordinate , ( or for that matter , an increasing coordinate and decreasing , taking into account the cosmological interpretation of section [ sec : cosmo ] ) can not have come in from the exterior region .equation ( [ gen - geod ] ) may be rewritten as if and , then the constant reduces to for a point particle with an initial velocity and initial time , then . if the particle is initially at rest then .an advantage of analyzing the interior region , not as a continuation of the exterior region , but as a manifold on its own , is a verification of the great difference existing between the geodesics of both regions .if one treats the interior solution as a cosmological solution , one may verify which type of universe one is dealing with , or which geodesics are analogous with those existing in our universe .  consider the geodesic equation given by where is an affine parameter defined along the geodesic .it is a simple matter of exercising some index gymnastics to verify the equivalence of the geodesic equation and the euler - lagrange equations ( [ euler - lagrange ] ) .now , the geodesic equation , eq .( [ geod - eq ] ) , for the metric ( [ intmetric ] ) may be written in the following form   considering the particular case of , and using the conserved quantities , the three primary integrals are given by which are identical to eqs .( [ p])-([line1 ] ) .( see ref . for an interesting analysis of radial geodesics confined under the schwarzschild horizon . )we shall next analyze null and timelike geodesics in some detail , and finally summarize the main results in tables [ t1 ] and [ t2 ] , respectively .equation ( [ line1 ] ) , for null geodesics , reduces to consider null geodesics along the , i.e. , with , so that we simply have where is an affine parameter defined along the geodesic .  for this case, the line element reduces to considering null geodesics , , i.e , , we have as solution +c  \ , ,    \label{z - null}\ ] ] where is a constant of integration .equation ( [ z - null ] ) is represented in fig .[ fig : null - z ] .direction , i.e. , with .see the text for details.,width=249 ]  note that traditionally the solution with corresponds to a black hole solution , either with an increasing or decreasing coordinate , i.e. , or , respectively .a white hole solution corresponds to , either with or .we also emphasize the importance of analyzing the interior solution separately , as in the literature the radial coordinate ( considered in the schwarzschild exterior ) is generally considered as a coordinate that measures distances , in the interior .it is usually treated as a temporal coordinate to note that decreases ( increases ) for an observer in a black hole ( white hole ) .it is also of interest to study the case of and .note that these are not circular orbits , as the coordinate can no longer be considered as a radial coordinate .equation ( [ pz ] ) provides , and thus eq .( [ nullline ] ) may be rewritten as the line element , for this particular case takes the form the null geodesic , , provides , which has the following solution or $ ] .one may also obtain the equivalent solution , given by the constant of integration has been chosen to provide for .note that for , then .for this case one verifies that a photon only traverses half - way around this particular universe ..summary of the equations of motion for null geodesics , considering the particular case of .see the text for details . [ cols= " < , < " , ]the eddington - finkelstein transformation is traditionally considered a transformation that permits the analysis of trajectories from .however , in a general manner , the inversion of the character of the coordinates is not manifest .therefore , to manifest this difference , we shall treat the eddington - finkelstein transformations directly from the interior metric ( [ intmetric ] ) .  for null geodesics along the ,( [ z - null ] ) provides the following solutions +c \,.\end{aligned}\ ] ] the solution with the negative sign shows that increases as , and decreases as ; from the solution with the positive sign , one may infer that increases as , and decreases as .consider now the following transformations   in the exterior region of the event horizon , solutions for are excluded , as one admits that the temporal coordinate increases . in the interior regiontwo distinct cases need to be separated , namely , for , which traditionally is denoted a black hole , and , a white hole .  taking into account the definition ,one may rewrite the metric ( [ intmetric ] ) as which is no longer singular at .now metric ( [ intmetricef ] ) may be simplified by introducing a null coordinate , denoted the advanced time parameter in analogy with the exterior solution so that the metric ( [ intmetric ] ) takes the form this is the line element of eddington - finkelstein for the advanced time parameter , which is regular at the instant .analyzing the specific case of , the metric ( [ intmetricef2 ] ) provides the following solutions recalling that , the above cases with have the following solutions these are plotted in fig .[ fig : edding - finkel ] , for different values of the constant .note that both solutions obey .a black hole solution corresponds to , and consequently ; and analogously , a white hole solution corresponds to and .  , and consequently ; and analogously , a white hole solution corresponds to and .,width=268 ]  applying an analogous procedure for the retarded temporal parameter , , constructed from , and consequently as is manifest from the line elements ( [ intmetricef2 ] ) and ( [ intmetricef3 ] ) , the metric coefficients are regular at .  for the case , the metric ( [ intmetricef3 ] ) provides the following relationships with the respective solutions these are plotted in fig .[ fig : edding - finkel2 ] , for different values of the constant .both solutions obey , with and corresponding to a black hole solution ; and and to a white hole solution , respectively .  , and consequently ; and analogously , a white hole solution corresponds to and .,width=268 ]consider the difference obtained from eqs .( [ def - v ] ) and ( [ def - w ] ) , given by from which one may obtain the following equalities substituting these expressions in eq .( [ intmetricef2 ] ) , one obtains   now , introducing the kruskal coordinates for the region ( ) , i.e. , substituting these expressions in eq .( [ def1 ] ) , we finally have   equations ( [ def - w])-([def - v ] ) may be rewritten as which substituting into metric ( [ intmetricef4 ] ) , we have the following   it is still possible to introduce the following transformations so that we have .the line element finally assumes the form   the new coordinates may be rewritten as     \nonumber  \\ & = & \left(1-\frac{t}{2\xi}\right)^{1/2}\exp\left(\frac{t}{4\xi}\right)\,\cosh\left(\frac{z}{4\xi}\right )            \label{def - t'2}\\ r_*&=&\frac{1}{2}\left[\exp\left(\frac{v'}{4\xi}\right ) -\exp\left(-\frac{w'}{4\xi}\right ) \right ]    \nonumber \\ & = & \left(1-\frac{t}{2\xi}\right)^{1/2}\exp\left(\frac{t}{4\xi}\right)\,\sinh\left(\frac{z}{4\xi}\right)\ , .\label{def - r'2}\end{aligned}\ ] ] these expressions may be written which is the equation for a hyperbole , and may also be expressed as which represent straight lines with .[ fig : diagram ] .the singularity at , written in terms of the new coordinates , is given by for , we have . for , we have , i.e. , , which implies . these relationships may be visualized in fig .[ fig : diagram ] .we have also added the exterior region , for comparison purposes ( see , for instance , ref . ) .    with this analysis at hand, one may consider the following motion of a timelike test particle , as viewed from an interior observer .the test particle starts its movement at the event , arriving at the surface of and , at event .after an excursion in the exterior region , the test particle re - enters into the interior region at event , corresponding to and , and finally ends up in the spacelike singularity at , at event .note an extremely curious feature of this movement , as viewed from an interior observer .the test particle exits the interior region , at the moment of complete contraction along the negative end of the direction , to reappear instantaneously at , at the positive side of the axis . according to the point of view of the interior observer ,no time has elapsed during the test particle s excursion in the exterior region .this analysis is analogous to the one outlined in ref .another curious feature , relatively to the interior observer is also worth mentioning : all infalling null or timelike particles enter into the interior at different places , but _ simultaneously _ at ., i.e. , at , and arrives at and , at event . it re - enters into the interior region at event , corresponding to and , ending up in the spacelike singularity at , at event . as viewed from an interior observer the test particle exits the interior region , at and , to reappear instantaneously at , at the positive side of the axis . according to the point of view of the interior observer ,no time has elapsed during the test particle s excursion in the exterior region.,width=307 ]the schwarzschild solution has played a fundamental conceptual role in general relativity , and beyond , for instance , regarding event horizons , spacetime singularities and aspects of quantum field theory in curved spacetimes . in this work ,we have provided a brief pedagogical review and further analyzed the interior schwarzschild solution .firstly , by deducing the interior metric by considering time - dependent metric fields , we have analyze the interior region , without the prejudices inherited from the exterior region . with this geometry at hand ,we have payed close attention to several respective cosmological interpretations , and addressed some of the difficulties associated to spacetimes singularities .secondly , we have deduced the conserved quantities of null and timelike geodesics , and discussed several particular cases in some detail .finally , we examined the eddington - finkelstein and kruskal coordinates directly from the interior solution .a black hole is believed to have formed from the gravitational collapse of a massive body .however , events occurring in the interior of the event horizon are not observable for an exterior observer , and one may argue that relatively to the latter , black holes are not relevant physical objects .although the event horizon exists for exterior observers , all events in the range are accessible to the interior observers .if one looks at the interior geometry as a continuation of the exterior static solution , one comes across some extremely interesting conceptual difficulties , that question the very concept of a black hole .for instance , while for the exterior observer , infalling particles end up at a central singularity at , from the interior point of view , the proper distance along the increases , showing the existence of a cigar - like singularity .the latter singularity is a spacelike hypersurface , and the test particles are not directed towards a privileged point , however , in order to not violate causality they are directed along a temporal direction from to .a curious behavior relatively to an interior observer is also verified , as all infalling particles crossing the event horizon , occur simultaneously at . in this context , the eddington - finkelstein and kruskal transformations do indeed solve the coordinate singularity at , but do not solve the problems associated with the inversion of the and coordinates .assuming that is a temporal coordinate for , also signifies giving it a determined direction and duration , i.e. , the black hole , or for that matter a white hole , possesses a finite coordinate temporal duration .however , the exterior geometry is static , and once created does not disappear .an interesting feature relatively to the interior geometry is the issue of proper distances .the proper distance between two particles at rest separated by a constant , decreases along the as coordinate time flows from to , and increases as coordinate time flows backwards from to . in counterpart ,the proper distance between two simultaneous events along a spatial trajectory with and , increases as varies from to , and decreases when the temporal coordinate runs backwards from to .another surprising result , considering the interior point of view , is that the trajectories of particles at rest are geodesics , contrary to the exterior where particles at rest are necessarily accelerated .this fact is due to the non - static character of the interior geometry .  in this work ,we have addressed some conceptual difficulties related to the notion of black holes .the solutions that do away with the interior singularity and the event horizon , although interesting in themselves , sweep the inherent conceptual difficulties of black holes under the rug . in concluding , we note that the interior structure of realistic black holes have not been satisfactorily determined , and are still open to considerable debate .p. crawford and i. tereno , `` generalized observers and velocity measurements in general relativity , '' gen .* 34 * , 2075 ( 2002 ) [ arxiv : gr - qc/0111073 ] . v. j. bolos ,`` lightlike simultaneity , comoving observers and distances in general relativity , '' journ .. phys . * 56 * , 813 - 829 ( 2006 ) [ arxiv : gr - qc/0501085 ] ; v. j. bolos , `` intrinsic definitions of ` relative velocity ' in general relativity , '' [ arxiv : gr - qc/0506032 ] .j. earman , `` tolerance for spacetime singularities , '' found .* 26 * , 623 - 640 ( 1996 ) .p. o. mazur and e. mottola , `` gravitational condensate stars : an alternative to black holes , '' [ arxiv : gr - qc/0109035 ] ; p. o. mazur and e. mottola , `` gravitational vacuum condensate stars , '' proc .111 , 9545 ( 2004 ) [ arxiv : gr - qc/0407075 ] ; g. chapline , e. hohlfeld , r. b. laughlin and d. i. santiago , `` quantum phase transitions and the breakdown of classical general relativity , '' int .a 18 3587 - 3590 ( 2003 ) [ arxiv : gr - qc/0012094 ] .m. visser and d. l. wiltshire , `` stable gravastars - an alternative to black holes ?, '' class .* 21 * 1135 ( 2004 ) [ arxiv : gr - qc/0310107 ] ; c. cattoen , t. faber and m. visser , `` gravastars must have anisotropic pressures , '' class .quant . grav .* 22 * 4189 - 4202 ( 2005 ) [ arxiv : gr - qc/0505137 ] ; n. bili , g. b. tupper and r. d. viollier , `` born - infeld phantom gravastars , '' jcap 0602 * 013 * ( 2006 ) [ arxiv : astro - ph/0503427 ] .i. dymnikova , `` vacuum nonsingular black hole , '' gen .* 24 * , 235 ( 1992 ) ;  i. dymnikova , `` spherically symmetric space - time with the regular de sitter center , '' int . j. mod .d 12 , 1015 - 1034 ( 2003 ) [ arxiv : gr - qc/0304110 ] ; i. dymnikova and e. galaktionov , `` stability of a vacuum nonsingular black hole , '' class .* 22 * , 2331 - 2358 ( 2005 ) [ arxiv : gr - qc/0409049 ] .g. chapline , `` dark energy stars , '' [ arxiv : astro - ph/0503200 ] ; f. s. n. lobo , `` stable dark energy stars , '' class .quant . grav .* 23 * , 1525 ( 2006 ) [ arxiv : astro - ph/0508115 ] .d. n. vollick , `` anisotropic born - infeld cosmologies , '' gen .. grav . * 35 * , 1511 ( 2003 ) [ arxiv : hep - th/0102187 ] ; d. n. vollick , `` black hole and cosmological space - times in born - infeld - einstein theory , '' [ arxiv : gr - qc/0601136 ] .v. v. kiselev , `` radial geodesics as a microscopic origin of black hole entropy .i : confined under the schwarzschild horizon , '' phys .d * 72 * 124011 ( 2005 ) [ arxiv : gr - qc/0412090 ] .r. m. wald , _ general relativity _, university of chicago press , chicago , 1984 .

Abstract: the schwarzschild solution has played a fundamental conceptual role in general relativity , and beyond , for instance , regarding event horizons , spacetime singularities and aspects of quantum field theory in curved spacetimes . however , one still encounters the existence of misconceptions and a certain ambiguity inherent in the schwarzschild solution in the literature . by taking into account the point of view of an observer in the interior of the event horizon , one verifies that new conceptual difficulties arise . in this work , besides providing a very brief pedagogical review , we further analyze the interior schwarzschild black hole solution . firstly , by deducing the interior metric by considering time - dependent metric coefficients , the interior region is analyzed without the prejudices inherited from the exterior geometry . we also pay close attention to several respective cosmological interpretations , and briefly address some of the difficulties associated to spacetime singularities . secondly , we deduce the conserved quantities of null and timelike geodesics , and discuss several particular cases in some detail . thirdly , we examine the eddington - finkelstein and kruskal coordinates directly from the interior solution . in concluding , it is important to emphasize that the interior structure of realistic black holes has not been satisfactorily determined , and is still open to considerable debate .