Delta distributions supported on quadratic O „ p , q … -invariant surfaces

Delta distributions supported on quadratic O „ p , q … -invariant surfaces

Ghislain R. Franssens^a兲

Belgian Institute for Space Aeronomy, Ringlaan 3, B-1180 Brussels, Belgium 共Received 12 February 2009; accepted 3 April 2009; published online 14 May 2009兲

An important subset of distributions encountered in physics is the set of multiplet delta distributions ␦c_y

共k兲, with support a quadratic O共p,q兲-invariant surface cy兵x苸Rⁿ:P共x兲=y其,n=p+q. The evaluation of these distributions for a general test function is not always sufficiently detailed in the classical literature, especially for those distributions that are defined as a regularization and/or when one needs their causal or anticausal version共forp= 1 orq= 1兲. This work intends to improve this situation by deriving explicit expressions for 具␦c_y

共k兲,␸典, ∀p,q苸Z₊, ∀k苸N,

∀y苸R, and∀␸苸D共Rⁿ兲, in a form suitable for practical applications. In addition, we also apply to these distributions a new approach to regularization. Distributions that need to be regularized are 共equivalently兲 defined as extensions of a partial distribution. This extension process reveals in a natural way that regularized mul- tiplet delta distributions are in general, uncountably multivalued. In the work of Gel’fand and Shilov关Generalized Functions共Academic, New York, 1964兲, Vol. 1兴 four types of multiplet delta distributions with support the null space c₀ were introduced: ␦1共k兲共P兲, ␦2共k兲共P兲 and ␦^共k兲共P₊兲, ␦^共k兲共P₋兲. Our regularization method ex- plains why this particular nonuniqueness was observed and further discloses the full extent of this nonuniqueness. ©2009 American Institute of Physics.

关DOI:10.1063/1.3124772兴

I. INTRODUCTION

Let p,q苸Z₊ and n=p+q. We revisit the definition of multiplet delta distributions ␦c_y 共k兲, ∀k 苸N, having as support a quadratic O共p,q兲-invariant surface cy兵x苸Rⁿ:P共x兲=y其, ∀y苸R, wherein P has the form 共5兲. This includes c₀, the null space of the ultrahyperbolic metric共6兲, which reduces to the共full兲light cone in the case of Minkowski space forp= 1 andq= 3.

Multiplet delta distributions are of fundamental importance in physics. For instance, the delta distribution␦c_yallows to calculate the integral of a共compactly supported兲function␸关actually of

␸/共2兩x兩兲, see共13兲兴over a mass hyperboloid, with massm=y^1/2, as具␦c_y,␸典. Further, the distribu- tion ␦c₀

共共n−2兲/2−l兲 is 共up to a multiplicative constant兲 a fundamental solution of the “iterated” ultra- hyperbolic wave equation 씲^lu=␦^{, for l}苸Z_{关1,共n−2兲/2兴}, in a higher-dimensional Minkowski space M^p,q, with odd number of time dimensionsp and odd number of space dimensions q, and when 4艋p+q共Ref.4, p. 281兲,共Huygens’ principle兲. More generally, the multiplet delta distribution␦c_y

共k兲

has as support de Sitter space dS^p−1,qify⬎0 or anti-de Sitter space AdS^p,q−1 ify⬍0.

The multiplet delta distributions concentrated at c₀, ␦c₀

共k兲, ∀k苸N, were introduced and dis- cussed in Ref.4共Chap. III, Sec. II A, p. 247兲and there denoted␦^共k兲共P兲. However, the expressions given there are not always of a sufficient explicitness to readily evaluate 具␦c_y

共k兲,␸典 in practical applications. For instance, it is not immediately obvious from the material in Ref. 4 how to evaluate these functionals in case they are defined as regularizations and/or one needs temporal

a兲Electronic mail: ghislain.franssens@aeronomy.be. URL: http://www.aeronomy.be/.

共 兲

50, 053512-1

0022-2488/2009/50共5兲/053512/31/$25.00 © 2009 American Institute of Physics

causal and anticausal共p= 1,q⬎1兲or spatial causal and anticausal共p⬎1,q= 1兲distributions. One of the purposes of this paper is to supply the missing details. The scope of the here presented treatment is also somewhat more general than Ref.4共Chapter III, Sec. II A兲as we also consider multiplet delta distributions having as support single and double hyperboloid sheets共i.e., the cases y⫽0兲. We derive explicit formulas to evaluate 具␦c_y

共k兲,␸典, ∀p,q苸Z₊, ∀k苸N, ∀y苸R and ∀␸

苸D共Rⁿ兲, including the temporal共ifp= 1兲and spatial共ifq= 1兲causal共or retarded, future, forward兲 and anticausal共or advanced, past, backward兲versions.

The distributions ␦c_y

共k兲 are most naturally defined as the pullback P^* of the one-dimensional delta distributions␦^共k兲 along the quadratic submersion P:Rⁿ\兵0其→Rsuch thatx哫y=P共x兲. The fact that the domain of the submersionPmust necessarily exclude the origin inRⁿemerges as a technical difficulty if one wants to define␦c共k_y兲 for all test functions in D共Rⁿ兲. This is the reason that, for certain values of p,q and k, one finds that the defining integrals for ␦c₀

共k兲 require a regularization. In Ref.4 共p. 249兲, an intriguing fact is observed in this respect. For even n and when共n− 2兲/2艋k, two different distributions are obtained with supportc₀, there labeled␦₁^共k兲共P兲 and ␦₂^共k兲共P兲. Moreover in Ref. 4 共p. 278兲 two more distinct delta distributions are introduced,

␦^共k兲共P+兲and␦^共k兲共P−兲, also having supportc₀. Any two of these four distributions differ from each other by a distribution having as support the origin兵0苸Rⁿ其 共Ref.4, p. 279兲. The second aim of this paper is to clarify this apparent nonuniqueness of certain multiplet delta distributions. We will show that, forn even and when共n− 2兲/2艋k, the distributions ␦c_y

共k兲 are, in general, uncountably multivalued. The key that fully unravels this nonuniqueness property is the more modern concept of the extension of a partial distribution, introduced by the author in Ref.2 and here recalled in Sec. II C. The four distributions␦₁^共k兲共P兲,␦₂^共k兲共P兲,␦^共k兲共P+兲and␦^共k兲共P−兲are then found to be just four particular extensions of a common unique partial distribution␦c₀

共k兲.

We also find that ifp= 1共q= 1兲and共n− 2兲/2艋k, the temporal共spatial兲causal and anticausal distributions are uncountably multivalued extensions, irrespective the parity ofn. This behavior of the causal and anticausal distributions differs from that of their time-symmetric共space-symmetric兲 counterparts, which are unique analytic continuations for n odd and uncountably multivalued extensions forneven. This appears to be a new result.

The outline of the paper is as follows.

共a兲 We first recall in Sec. II the definition of the pullback of a distribution along a scalar submersion and apply this to the quadratic function P.

共b兲 We consider in Sec. III the distribution ␦c_y for generaly苸R.

共c兲 The distribution␦c_yis fundamental to define the pullback of any distribution alongP. We use

␦c_yto calculate the particular distributions ␦c_y

共k兲,∀k苸N, in Sec. IV.

共d兲 We discuss in Sec. V, as an interesting and practical example, the temporal causal 共⫹兲and anticausal共⫺兲distributions共␦_c

0 t⫾

共1兲兲efor p= 1 andq= 3.

Finally a note on terminology. We say that a distribution f苸D⬘共X兲, having supportU債X, is definedfor test functions inD共X兲 关withD⬘^共X兲the continuous dual space ofD共X兲兴, isbasedonX 共called the base space off兲and isconcentratedatU. Further, we use the notation and definitions as stated in Ref.2. For convenience we recall here the 1-symbol, 1p1 ifpis true, else 1p0, the parity symbols,em1m苸Z_eandom1m苸Z_o 共Ze,oeven or odd integers兲, and the step functions 1_⫾ 共characteristic functions ofR_⫾兲.

II. PULLBACK TO RⁿOF A DISTRIBUTION BASED ON R

A. Pullback along a scalar submersion

Denote by具,典dRthe de Rham pairing, of a chain and a form, and by具,典the Schwartz pairing, of a generalized function and a test function.

The de Rham pairing 具,典dR of anm-chaincy債X債Rⁿ and an m-form ␸␻T is defined as the integral of ␸␻Tovercy,共Ref. 1, p. 217兲. Herein is␻Tthe Leray form of the chaincy, such that

␻X=dT∧␻T, with␻Xthe volume form onX.

For any regular distribution f苸D⬘共X兲, the Schwartz pairing is defined as

具f,␸典具X,f␸␻X典dR. 共1兲

Definition 1:Let n苸N: 2艋n,X債Rⁿ,Y=Rand␦y苸D⬘^{共Y兲with具}␦y,␺典␺共y兲,∀␺苸D共Y兲.

Let f苸D⬘^{共Y兲 and T:X→}Y such that x哫y=T共x兲 be a C^⬁ function with 共dT兲共x兲⫽0, ∀x 苸c_y兵x苸X:T共x兲=y其,and ∀y苸suppf.The pullback T^*f of f along T is defined∀␸苸D共X兲as

具T^*f,␸典具f,⌺T␸典, 共2兲

with

共⌺T␸兲共y兲具T^*␦y,␸典. 共3兲

The distribution␦c_yT^*␦y苸D⬘^共X兲represents a delta distribution having as support the level set surfacecyofT, with level parametery, and is defined as共Ref.3, p. 85, and Ref.1, p. 438兲

具␦c_y,␸典具cy,␸␻T典dR. 共4兲

It is clear from共2兲and共3兲that the pullback␦c_yof the one-dimensional distribution␦y, defined by 共4兲, is fundamental to calculate the pullbackT^*f of any distribution f along T.

The condition ondTin Definition 1 is necessary and sufficient for the Leray form␻Tto exist oncy. Moreover, although␻X=dT∧␻Tdoes not specify␻Tuniquely in a neighborhood ofcy,␻T

is unique oncy共Ref.4, pp. 220 and 221兲. The Leray form␻Tis the natural surface form oncy. We can not speak of the delta distribution with supportcysince the pullbackT^*␦y, as defined above, depends on the equation used to represent the surfacecy, through the Leray form␻T共Ref.

4, p. 222, and Ref. 1, p. 439兲. For this reason, it might sometimes be appropriate to use a more explicit notation for␦c_y, that displays the form of the equation used to describecy, such as␦共T共x兲−y兲. It is shown in, e.g., Ref.3, p. 82, Theorem 7.2.1, that, under the conditions given in Definition 1,⌺T␸苸D共Y兲,T^*f苸D⬘^{共X兲, and T*}is a sequentially continuous linear operator.

B. The quadratic submersionP

Let p,q苸Z₊ withn=p+q,共x^␮兲x共xt苸R^p,xs苸R^q兲苸Rⁿ and兩 兩 the Euclidean norm on Rⁿ. Define a functionP:X=Rⁿ\兵0其→Y=Rsuch thatx哫y=P共x兲with共implicit summation over 1 ton兲

P共x兲␩␮␯x^␮x^␯=兩xt兩²−兩xs兩², 共5兲 wherein

共6兲

We havedP= 2␩␮␯x^␮dx^␯⫽0,∀x苸X, hencedP is everywhere surjective and Pis a submer- sion. Define the level setscy兵x苸Rⁿ:P共x兲=y其,∀y苸R. In particular,c₀is the null space共or null cone ifp= 1 orq= 1兲with respect to the origin. Notice that our definition ofc₀includes the origin, soc₀X andc₀ is not a manifold,共Ref.1, p. 113兲.

1. Volume form

The natural volume form on X most easily follows from Hodge’s star operator

*

^{as ␻X}

=

*

^{1 or}

␻X=⑀1¯ndx¹∧ ¯ ∧dxⁿ, 共7兲 with⑀then-covariant Levi–Civita pseudotensor field, involving the metric onX. The volume form

␻X is an antisymmetric tensor density field of order n and weight 1. A tensor density field of weight 1 is also called a pseudotensor field. An antisymmetric tensor density field of weight 1 is usually called an odd共or twisted or pseudo-兲form.

The classical volume elementdVis the infinitesimal Euclidean measure corresponding to the volume form␻X onXaccording to,∀␸苸D共X兲,

具X,␸␻X典dR

冕

X

␸共x兲dV共x兲. 共8兲

2. Leray form

Introduce the interior product ∨:⌫共T^*X兲⫻⌫共T^*X兲→R such that 共␣,␤兲哫␣∨␤=␩^␮␯␣_␮␤_␯, with 关␩^␮␯兴关␩␮␯兴⁻¹. The Leray form ␻P follows from the following property of Hodge’s star operator,∀␣^,␤苸⌫共T^*X兲:

␣∧共ⴱ␤兲=共ⴱ␣兲∧␤=共␣∨␤兲␻X, 共9兲 with ∧ the exterior product. Let ␣=dP,

*

^␤=␻P, and impose that dP∨共

*

^{−1␻P兲= 1. Since}

dP∨dP= 4P, we get ␻P=

*

^{共dP/4P兲or}

␻P= 1

1!共n− 1兲!⑀_{␮1␮2¯␮n}^x␮1

2P共dx^␮2∧ ¯ ∧dx^␮n兲. 共10兲 The Leray form␻Pis also an odd form.

Let · :⌫共T^*X兲⫻⌫共T^*X兲→R such that 共␣,␤兲哫␣·␤=␦^␬␭␣_␬␤_␭stand for the Euclidean inner product on⌫共T^*X兲, so兩␣兩=

冑

_␣·␣. The outward unit normal 1-formn苸⌫共T^*X兲such that兩n兩= 1 is defined as

n dP

兩dP兩=2␩␮␯x^␮dx^␯

2兩x兩 . 共11兲

The classical surface elementdSis the infinitesimal Euclidean measure corresponding to the Leray form␻Ponc_yaccording to共Ref.1, p. 439兲,∀␸苸D共X兲,

冓

^{cy,␸1!共n1− 1兲!⑀␮1␮2¯␮n共dx␮2∧ ¯ ∧dx␮n兲}

冔

^dR

^冕

^{cyn␮1共x兲␸共x兲dS共x兲. 共12兲}

Substituting in共12兲the expressions for the normal componentsn_␮

1, read off from共11兲, and using expression共10兲for␻P, we obtain for共4兲the integral form,∀y苸R,

共⌺P␸兲共y兲=1 2

冕

c_y

␸共x兲

兩x兩 dS共x兲. 共13兲

3. Pullback along P

The following two cases can occur.

共i兲 Af苸D⬘^共R兲 for which 0苸suppf has a pullbackP^*f苸D⬘^{共Rn\兵0其兲.}

共ii兲 Af苸D⬘^共R兲 for which 0苸suppf has a pullback P^*f苸D⬘^共Rn兲, since it is sufficient that dP⫽0 on all level sets ofPcorresponding to the support of f.

C. Extension of a partial distribution

In distribution theory, one is often forced to impose restrictions on the test functions to which certain functionals can be applied. An example of this is case共i兲above. In this section we describe a general procedure which can remove such restrictions. We will use this procedure in the subse- quent sections to construct multiplet delta distributions, which are defined for all test functions in D共Rⁿ兲. This means that our results will be applicable to test functions having a support that may intersect with the origin兵0苸Rⁿ其. If one only wants to apply a multiplet delta distribution to test functions, vanishing in a neighborhood of the origin, then extensions are obviously unnecessary.

Definition 2:LetDr傺Dand denote byD⬘r傻D⬘the continuous dual space ofDr.Elements of Dr⬘that are only defined onDrare calledpartial distributions defined onDr.

Example: P^*f in case共i兲above is a partial distribution defined onD共Rⁿ\兵0其兲傺D共Rⁿ兲.

A distribution, by definition, is a linear 共sequentially兲 continuous functional defined on the whole of D.^8,9 We want to extend a partial distribution, defined on Dr, so that it becomes a distribution. A natural way to do this is as follows.

Since the space of test functionsDis locally convex共Ref.7, p. 9兲, the Hahn–Banach theorem guarantees that a continuous linear functional f, defined on aDr傺D共i.e., a partial distribution兲, can be extended to a continuous linear functionalfedefined on the whole ofD共i.e., a distribution兲, and that both coincide onDr. The subset ofDr⬘^{that maps}Drto zero is called the annihilator ofDr, denotedDr⬘^⬜. Clearly, any two extensionsf_e,1andf_e,2off can differ by an element inDr⬘^{⬜. If}Dr

is dense inD, then the extension is unique.

Notice that this method, when applied toDr=D傺S, shows that any tempered distribution in S⬘^共Ref.8, Vol. II, pp. 89–104兲can be regarded as the extension of a distribution inD⬘^傻S⬘^{. Since} Dis a dense subspace ofS共Ref.9, p. 101兲, any tempered distribution, regarded as an extension, is unique.

The Hahn–Banach theorem does not say how an extension can be constructed. A natural extension process, however, is the one used in the distribution literature for the regularization of divergent integrals共Ref.4, pp. 10 and 45兲. This procedure basically consists in the construction of a projection operatorT:D→Drand to replace an integral, convergent∀␺苸Drbut divergent for test functions␸苸D\Dr共e.g., in one dimension兲,

冕

−⬁ +⬁

f共x兲␸共x兲dx, 共14兲

by the integral

冕

−⬁ +⬁

f共x兲共T␸兲共x兲dx, 共15兲

which now by construction is convergent∀␸苸D. The integral共15兲is then said to be a regular- ization of the integral共14兲. Equivalently, we could say that共15兲defines a distribution which is an extension of the partial distribution defined by共14兲. It is important to notice, however, that this method is by no means unique, since infinitely many projection operatorsT can be constructed, mappingDtoDr.

For distributions that depend on a complex parameterz, such asx_⫾^z 共Ref.4, p. 48, and Ref.5, p. 84兲, this procedure can be applied to its Laurent series about a polez= −l苸Z₋. Any共associated homogeneous兲extensionx_⫾^z_,eis then obtained in the form

x_⫾^−l_,e=x^−l_⫾,0+c␦^共l−1兲, 共16兲 withx_⫾,0^−l the analytic finite part共i.e., the zeroth term of the Laurent series兲at the polez= −land c苸Cis arbitrary. The distribution x_⫾,0^−l is given by关Ref.2, Eq.共108兲兴

具x_⫾,0^−l ,␸典=

冕

_−⬁^{+⬁共兩x兩−l1⫾共x兲兲共Tl−1,0␸兲共x兲dx, 共17兲}

wherein the projection operator is defined as

共T_p,q␸兲共x兲␸共x兲−

兺

l=0 p+q

␸^共l兲共0兲x^l

l!共1_l⬍p+ 1_p艋l1₊共1 −x²兲兲. 共18兲 Any so constructed extensionx_⫾^−l_,eis homogeneous of degree −land has first order of association.

More details of this procedure can be found in Ref.2.

An advantage of the concept “extension of a partial distribution” is that it clearly reveals why many singular distributions are generally nonunique. The classical definition of such a distribution is, almost always, given as a particular regularization of an integral, and this approach has largely obscured the intrinsic nonuniqueness of this process.

III. THE DELTA DISTRIBUTION␦c_y

A. Summary of results

The distribution

␦c_yP^*␦y 共19兲

represents the delta distribution with supp共␦c_y兲=cy, relative to the equationP共x兲=ywhich defines cy, and is given by共4兲.

In this section, we will obtain the following results.

共i兲 Fory⫽0,␦c_y always exists as a distribution inD⬘^共Rn兲. For example, forp= 1 andy=m²

⬎0, ␦c_y is the delta distribution concentrated at the 共double sheeted兲 timelike mass m hyperboloid P共x兲=m².

共ii兲 Fory= 0, we will find that␦c₀is apartial distributiondefined onD共R²\兵0其兲ifp= 1 =qand else is a distribution inD⬘共Rⁿ兲. In the former case, our extension process generates a 共in general, nonunique兲 distribution 共␦c₀兲e苸D⬘共R²兲, called delta distribution concentrated at the共full兲null cone P共x兲= 0, and which coincides with the partial delta distribution␦c₀ in the sense that具␦c₀,␺典=具共␦c₀兲e,␺典,∀␺苸D共R²\兵0其兲. For example, forp= 1 andq⬎1,␦c₀is the共unique兲delta distribution concentrated at the null coneP共x兲= 0.

Deriving an explicit expression for the functional value 具␦c₀,␸典 requires distinguishing the following four cases.

B. Case p>1 and q>1

For distributions concentrated at an O共p,q兲-invariant subspace of R^p,q, one can typically derive two forms for⌺P␸ⁱⁿ共2兲, hereafter labeled共a兲and共b兲. Form共a兲is obtained by integrating over the radius of the temporal section, ␳t, and form 共b兲 by integrating over the radius of the spatial section,␳s. This leads to two equivalent expressions as fory⫽0 in the following.

Proposition 3: The delta distribution ␦c_y, concentrated at P共x兲=y, is for p⬎1, q⬎1 and

∀␸苸D共Rⁿ兲 given by the following expressions.

共i兲 For y⫽0,

具␦c_y,␸典=1 2

冕

0

+⬁

1₊共␳t 2−y兲␳t

p−1共

冑

␳t

2−y兲^q−2␺^S,S共␳t,

冑

␳t

2−y兲d␳t 共20兲

=1 2

冕

0

+⬁

1₊共␳s2+y兲共

冑

_␳s²+y兲^p−2␳q−1s ␺^S,S共

冑

_␳s²+y,␳s兲d␳s. 共21兲 共ii兲 For y= 0,

具␦c₀,␸典=1 2

冕

0

+⬁

␳ⁿ⁻³␺^S,S共␳^,␳兲d␳^. 共22兲 Herein is␺^S,S:R²→Rsuch that共t,s兲哫␺^S,S共t,s兲,with

␺^S,S共t,s兲具St

p−1⫻S_s^q−1,␸共t␻t,s␻s兲共␻S t

p−1∧␻S

s

q−1兲典dR. 共23兲

Proof: Introduce bispherical coordinates共␳t苸R₊,␻t苸S_t^p−1;␳s苸R₊,␻s苸S_s^q−1兲inX. Let␻S t p−1

and␻S s

q−1be the volume forms on the unit spheresS_t^p−1andS_s^q−1 of the time and space sections, respectively. Now, P=␳t2−␳s2. The volume form ␻X becomes 共suppressing the Levi–Civita pseudotensor field⑀which numerically amounts to a factor 1兲

␻X=共␳t

p−1d␳t∧␻S t p−1兲∧共␳s

q−1d␳s∧␻S s q−1兲.

The mapping共23兲from␸→␺^S,Sis sequentially continuous and␺^S,S苸D共R²兲,∀␸苸D共Rⁿ兲. More- over,␺^S,S is even intand ins共Ref. 4, p. 250兲.

Form (a). In terms of the coordinates P=y苸R,␳t苸R₊, and ␻t,␻s, we get for the volume form

␻X=␳t

p−1d␳t∧␻S t

p−1∧¹₂共

冑

_␳t²−y兲^q−2dy∧␻S s

q−1, 共24兲

with the argument of the square root restricted to positive values. The Leray form is read off as

␻P= ¹₂␳tp−1共

冑

␳t2−y兲^q−2d␳t∧␻S t

p−1∧␻S

s

q−1. 共25兲

Notwithstanding␻Xand␻P are odd forms, any permutation of the basis forms never generates a change in sign because this transformation involves the modulus of the Jacobian determinant of the permutation. For the same reason, any square root in共24兲and共25兲must be taken as positive.

Substituting共25兲in共4兲and using共3兲gives 共⌺P␸兲共y兲=1

2

冕

0 +⬁

1₊共␳t 2−y兲␳t

p−1共

冑

_␳t²−y兲^q−2␺^S,S共␳t,

冑

_␳t²−y兲d␳t. 共26兲 Form (b). In terms of the coordinates P=y苸R, ␳s苸R₊, and ␻t,␻s, we get for the volume form

␻X=¹₂共

冑

_␳s²+y兲^p−2dy∧␳s

q−1d␳s∧␻S t

p−1∧␻S

s

q−1, 共27兲

with the argument of the square root again restricted to positive values, and for the Leray form

␻P=¹₂共

冑

␳s2+y兲^p−2␳sq−1d␳s∧␻S t

p−1∧␻S

s

q−1. 共28兲

Substituting共28兲in共4兲gives

共⌺P␸兲共y兲=1 2

冕

0

+⬁

1₊共␳s

2+y兲共

冑

_␳s²+y兲^p−2␳s

q−1␺^S,S共

冑

_␳s²+y,␳s兲d␳s. 共29兲 共i兲 Fory⫽0, either共26兲or共29兲defines⌺P␸^,∀␸苸D共Rⁿ兲. Both共26兲and共29兲as well as the mapping共23兲are sequentially continuous linear operators, acting on␸^{. Then,}␦c_y, defined by共2兲and either共26兲or共29兲together with共23兲, is a distribution based onRⁿ.

共ii兲 Ify= 0 we get, from either 共26兲or共29兲, 共⌺P␸兲共0兲=1

2

冕

0 +⬁

␳ⁿ⁻³␺^S,S共␳^,␳兲d␳^. 共30兲 Reordering the limit y→0 and the integration in 共26兲 and 共29兲 is permitted since the integrands are jointly continuous in y and the integration variable. Hence, 共⌺P␸兲共0兲

=具x₊ⁿ⁻³,␸1典with␸1共x兲¹₂␺^S,S共x,x兲and␸1苸D共R兲. Sincen艌4,具x₊ⁿ⁻³,␸1典exists. Then␦c₀, defined by共2兲together with共30兲and共23兲, is a distribution based onRⁿdue to the sequen- tial continuity of the mapping共23兲 and the sequential continuity of the one-dimensional

distributionx₊ⁿ⁻³. 䊐

Expressions共26兲and共29兲are easily shown to be equivalent. However, we will see in the next section that continuing with both forms 共a兲 and共b兲 is not superfluous, as they can give rise to different regularizations for certain multiplet delta distributions共a fact also observed in Ref.4, p.

251兲. This is also made explicit in Sec. V.

From 共22兲readily follows共with 1_共␻兲 the 1-distribution onS^m−1兲

␦c₀=¹₂共␦共t−s兲丢s₊ⁿ⁻³丢共1_共␻

t兲丢1_共␻

s兲兲兲, 共31兲

=¹₂共t₊ⁿ⁻³丢␦共s−t兲丢共1_共␻t兲丢1_共␻

s兲兲兲. 共32兲

C. Case p = 1 and q>1

Proposition 4: The delta distribution ␦c_y, concentrated at P共x兲=y, is for p= 1, q⬎1 and

∀␸苸D共Rⁿ兲 given by the following expressions.

(i) For y⫽0,

具␦c_y,␸典=1 2

冕

−⬁

+⬁

1₊共t²−y兲共

冑

t²−y兲^q−2␺^−,S共t,

冑

t²−y兲dt, 共33兲

=1 2

冕

0

+⬁

1₊共␳s2+y兲 ␳s q−1

冑

_␳s²+y⌿s共

冑

␳s2+y,␳s兲d␳s, 共34兲 with⌿s defined as

⌿s共t,␳s兲␺^−,S共+t,␳s兲+␺^−,S共−t,␳s兲. 共35兲 (ii) For y= 0,

具␦c₀,␸典=1 2

冕

0

+⬁␳^q−2␺^−,S共+␳^,␳兲d␳⁺¹ 2

冕

0

+⬁␳^q−2␺^−,S共−␳^,␳兲d␳^. 共36兲

Herein is␺^−,S:R2→Rsuch that共t,s兲哫␺^−,S共t,s兲,given by

␺^−,S共t,s兲具S^q−1_s ,␸共t,s␻s兲␻S s

q−1典dR. 共37兲

Proof: Introduce spherical coordinates共␳s苸R₊,␻s苸S_s^q−1兲 for the spatial section ofX. Now, P=共x¹兲²−␳s2and the volume form␻X is␻X=dx¹∧␳sq−1d␳s∧␻S

s q−1.

Form (a). Introduce the change in variables 共x¹,␳s兲→共t=x¹,y=共x¹兲²−␳s2兲 having Jacobian determinant −2

冑

t²−y. In terms of the coordinates t苸R, P=y苸R, and ␻s, the volume form becomes

␻X=¹₂共

冑

t²−y兲^q−2dy∧dt∧␻S s

q−1, 共38兲

with the argument of the square root restricted to positive values, and the Leray form is

␻P=¹₂共

冑

t²−y兲^q−2dt∧␻S s

q−1. 共39兲

Substituting共39兲in共4兲gives 共⌺P␸兲共y兲=1

2

冕

_−⬁^{+⬁1+共t2−y兲共}

^冑

^{t2−y兲q−2␺−,S共t,}

^冑

^{t2−y兲dt. 共40兲}

The mapping共37兲from␸→␺^−,Sis sequentially continuous and␺^−,S苸D共R²兲,∀␸苸D共Rⁿ兲. More- over,␺^−,Sis even ins共Ref. 4, p. 250兲.

Form (b). Introduce the change in variables共x¹,␳s兲→共y=共x¹兲²−␳s

2,␳s兲 with Jacobian deter- minant 2共␳s

2+y兲^1/2. In terms of the coordinatesP=y苸R,␳s苸R₊, and ␻s, the volume form is

␻X=1 2

冑

_␳s²1+ydy∧␳s

q−1d␳s∧␻S s

q−1, 共41兲

with the argument of the square root again restricted to positive values, and the Leray form is

␻P= 1 2

␳sq−1

冑

_␳s²+yd␳s∧␻S s

q−1. 共42兲

Substituting共42兲in共4兲gives共sincex¹苸R兲 共⌺P␸兲共y兲=1

2

冕

0 +⬁

1₊共␳s

2+y兲 ␳s q−1

冑

_␳s²+y⌿s共

冑

_␳s²+y,␳s兲d␳s. 共43兲 共i兲 Fory⫽0, either共40兲or共43兲defines⌺P␸^,∀␸苸D共Rⁿ兲. Then共2兲, together with either共40兲

or共43兲, defines␦c_yas a distribution based onRⁿ. 共ii兲 Ify= 0 we get, from either 共40兲or共43兲,

共⌺P␸兲共0兲= 1 2

冕

0

+⬁␳^q−2␺^−,S共+␳^,␳兲d␳^{+ 1} 2

冕

0

+⬁␳^q−2␺^−,S共−␳^,␳兲d␳^. 共44兲

Hence,共⌺P␸兲共0兲=具x^q−2₊ ,␸1典with␸1共x兲₂¹⌿s共x,x兲and␸1苸D共R兲,∀␸苸D共Rⁿ兲. Sinceq艌2, 具x₊^q−2,␸1典exists. Then␦c₀, defined by共2兲together with共44兲, is a distribution based on Rⁿ. 䊐 Equation共36兲corrects Ref.4, p. 252, Eq.共14⬘^兲withk= 0, where the second integral has the wrong sign due to not applying the modulus to the Jacobian determinant when transforming from 共x¹,␳s兲→共y,␳s兲.

The temporal future and past hyperboloid sheets共or ify= 0, half null cones兲with respect to the origin are the respective sets,∀y苸R,

c_y^t+兵x苸Rⁿ:共x¹兲²−␳s2=y,0艋x¹其, 共45兲

c_y^t−兵x苸Rⁿ:共x¹兲²−␳s

2=y,x¹艋0其. 共46兲

If y⬍0, 共45兲represents the 0艋x¹ part of a hyperboloid of one sheet, while 共46兲 represents its x¹艋0 part. If y⬎0,共45兲 represents the 0艋x¹ sheet of a hyperboloid of two sheets, while 共46兲 represents itsx¹艋0 sheet.

Corollary 5:

(i) Let y⫽0. The temporal causal delta distribution ␦c y

t+, concentrated at a future timelike mass hyperboloid sheet c_y^t+, and temporal anticausal delta distribution␦c

y

t−, concentrated at a past timelike mass hyperboloid sheet c^t−_y, are given by

具␦c y t+,␸典= 1

2

冕

0 +⬁

1₊共t²−y兲共

冑

t²−y兲^q−2␺^−,S共t,

冑

t²−y兲dt, 共47兲

=1 2

冕

0

+⬁

1₊共␳s2+y兲 ␳s q−1

冑

␳s2+y␺^−,S共+

冑

_␳s²+y,␳s兲d␳s, 共48兲 and

具␦c y t−,␸典=1

2

冕

_−⬁^{0 1+共t2−y兲共}

^冑

^{t2−y兲q−2␺−,S共t,}

^冑

^{t2−y兲dt, 共49兲}

=1 2

冕

0

+⬁

1₊共␳s2+y兲 ␳s q−1

冑

_␳s²+y␺^−,S共−

冑

␳2s+y,␳s兲d␳s. 共50兲 (ii) Let y= 0. The temporal causal delta distribution␦c

0

t+, concentrated at the future half null cone c₀^t+, and temporal anticausal delta distribution␦c₀^t−, concentrated at the past half null cone c₀^t−, are given by

具␦c 0 t+,␸典= 1

2

冕

0 +⬁

␳^q−2␺^−,S共+␳^,␳兲d␳^, 共51兲

具␦c 0 t−,␸典= 1

2

冕

0 +⬁

␳^q−2␺^−,S共−␳^,␳兲d␳^. 共52兲 Proof: The delta distribution ␦c

y t+ 共␦c

y

t−兲 is obtained by restricting to 0艋x¹ 共x¹艋0兲 and by integrating in共33兲from 0→+⬁共−⬁→0兲. Equivalently, we only use the first共second兲integral in

共34兲. 䊐

From 共51兲and共52兲readily follows that

␦c 0

t⫾=₂¹共␦_共t−共⫾s兲兲丢s₊^q−2丢1_共␻

s兲兲. 共53兲

D. Case p>1 and q = 1

Proposition 6: The delta distribution ␦c_y, concentrated at P共x兲=y, is for p⬎1, q= 1 and

∀␸苸D共Rⁿ兲 given by the following expressions.

(i) For y⫽0,

具␦c_y,␸典=1 2

冕

0

+⬁

1₊共␳t

2−y兲 ␳tp−1

冑

_␳t²−y⌿t共␳t,

冑

_␳t²−y兲d␳t, 共54兲

=1 2

冕

−⬁

+⬁

1₊共s²+y兲共

冑

s²+y兲^p−2␺^S,−共

冑

s²+y,s兲ds, 共55兲 with⌿tdefined as

⌿t共␳t,s兲␺^S,−共␳t, +s兲+␺^S,−共␳t,−s兲. 共56兲 (ii) For y= 0,

具␦c₀,␸典=1 2

冕

0

+⬁␳^p−2␺^S,−共␳^{, +}␳兲d␳⁺¹ 2

冕

0

+⬁␳^p−2␺^S,−共␳^,−␳兲d␳^. 共57兲

Herein is␺^S,−:R2→Rsuch that共t,s兲哫␺^S,−共t,s兲,given by

␺^S,−共t,s兲具Stp−1,␸共t␻t,s兲␻S_t^p−1典dR. 共58兲 Proof: The proof ismutatis mutandis similar to the proof of Proposition 4. Form共a兲 now becomes

共⌺P␸兲共y兲=1 2

冕

0

+⬁

1₊共␳t2−y兲 ␳t p−1

冑

_␳t²−y⌿t共␳t,

冑

␳t2−y兲d␳t, 共59兲 while form共b兲is

共⌺P␸兲共y兲=1

2

冕

_−⬁^{+⬁1+共s2+y兲共}

^冑

^{s2+y兲p−2␺S,−共}

^冑

^{s2+y,s兲ds. 共60兲}

共i兲 Fory⫽0,␦c_y, defined by共2兲together with either共59兲and共60兲, is a distribution based on Rⁿ.

共ii兲 Ify= 0 we get, from either 共59兲and共60兲, 共⌺P␸兲共0兲=1

2

冕

0

+⬁␳^p−2␺^S,−共␳^{, +}␳兲d␳⁺¹ 2

冕

0

+⬁␳^p−2␺^S,−共␳^,−␳兲d␳^. 共61兲

Again␦c₀, defined by共2兲together with共61兲, is a distribution based on Rⁿ. 䊐 Equation共61兲corrects Ref.4, p. 252, Eq.共15⬘兲withk= 0, where the second integral has the wrong sign due to not applying the modulus to the Jacobian determinant when transforming from 共␳t,xⁿ兲→共␳t,y兲.

The spatial forward and backward hyperboloid sheets共or ify= 0, half null cones兲with respect to the origin are the respective sets,∀y苸R,

c_y^s+兵x苸Rⁿ:␳t

2−共xⁿ兲²=y,0艋xⁿ其, 共62兲

c_y^s−兵x苸Rⁿ:␳t

2−共xⁿ兲²=y,xⁿ艋0其. 共63兲

If y⬎0, 共62兲represents the 0艋xⁿ part of a hyperboloid of one sheet, while 共63兲 represents its xⁿ艋0 part. If y⬍0,共62兲 represents the 0艋xⁿ sheet of a hyperboloid of two sheets, while 共63兲 represents itsxⁿ艋0 sheet.

Corollary 7:

(i) Let y⫽0. The spatial causal delta distribution ␦c y

s+, concentrated at a forward spacelike mass hyperboloid sheet c_y^s+, and spatial anticausal delta distribution␦c_y^s−, concentrated at a backward spacelike mass hyperboloid sheet c_y^s+, are given by

具␦c y s+,␸典=1

2

冕

0 +⬁

1₊共␳t

2−y兲 ␳tp−1

冑

_␳t²−y␺^S,−共␳t, +

冑

_␳t²−y兲d␳t, 共64兲

=1 2

冕

0

+⬁

1₊共s²+y兲共

冑

s²+y兲^p−2␺^S,−共

冑

s²+y,s兲ds, 共65兲 and

具␦c y s−,␸典= 1

2

冕

0 +⬁

1₊共␳t

2−y兲 ␳t p−1

冑

_␳t²−y␺^S,−共␳t,−

冑

_␳t²−y兲d␳t, 共66兲

=1

2

冕

_−⬁^{0 1+共s2+y兲共}

^冑

^{s2+y兲p−2␺S,−共}

^冑

^{s2+y,s兲ds. 共67兲}

(ii) Let y= 0. The spatial causal delta distribution ␦c 0

s+, concentrated at the forward half null cone c₀^s+, and spatial anticausal delta distribution␦c₀^s−, concentrated at the backward half null cone c₀^s−, are given by

具␦c 0 s+,␸典=1

2

冕

0

+⬁␳^p−2␺^S,−共␳^{, +}␳兲d␳^, 共68兲

具␦c 0 s−,␸典=1

2

冕

0 +⬁

␳^p−2␺^S,−共␳^,−␳兲d␳^. 共69兲 Proof: The delta distribution␦c_y^s+共␦c_y^s−兲 is obtained by restricting to 0艋xⁿ 共xⁿ艋0兲and inte- grating in 共55兲 from 0→+⬁ 共−⬁→0兲. Equivalently, we only use the first 共second兲 integral in

共54兲. 䊐

From 共68兲and共69兲readily follows that

␦c 0

s⫾=¹₂共t₊^p−2丢␦_共s−共⫾t兲兲丢1_共␻

t兲兲. 共70兲

E. Case p = 1 and q = 1

Proposition 8: The delta distribution ␦c_y, concentrated at P共x兲=y, is for p= 1 =q and ∀␸ 苸D共Rⁿ兲given by the following expressions.

(i) For y⫽0,

具␦c_y,␸典=1 2

冕

−⬁

+⬁

1₊共t²−y兲␸共t,

冑

t²−y兲

冑

t²−y dt+1 2

冕

−⬁

+⬁

1₊共t²−y兲␸共t,−

冑

t²−y兲

冑

t²−y dt, 共71兲

=1

2

冕

_−⬁^{+⬁1+共s2+y兲␸共}

^冑 冑

^ss²²⁺+^y,s兲y ds+1

2

冕

_−⬁^{+⬁1+共s2+y兲␸共−}

冑 ^冑

s^s22+⁺y^y,s兲ds.

共72兲 (ii) For y= 0,