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Static interactions and stability of matter in Rindler space

# Static interactions and stability of matter in Rindler space

F. Lenz    K. Ohta    K. Yazaki Institute for Theoretical Physics III
University of Erlangen-Nürnberg
Staudtstrasse 7, 91058 Erlangen, Germany

Institute of Physics
University of Tokyo
Komaba, Tokyo 153, Japan

Hashimoto Mathematical Physics Laboratory
Nishina Center, RIKEN
Wako, Saitama 351-0198, Japan

Yukawa Institute for Theoretical Physics,
Kyoto University
Kyoto 606-8502, Japan
December 15, 2010
###### Abstract

Dynamical issues associated with quantum fields in Rindler space are addressed in a study of the interaction between two sources at rest generated by the exchange of scalar particles, photons and gravitons. These static interaction energies in Rindler space are shown to be scale invariant, complex quantities. The imaginary part will be seen to have its quantum mechanical origin in the presence of an infinity of zero modes in uniformly accelerated frames which in turn are related to the radiation observed in inertial frames. The impact of a uniform acceleration on the stability of matter and the properties of particles is discussed and estimates are presented of the instability of hydrogen atoms when approaching the horizon.

###### pacs:
04.62.+v, 11.15.-q, 11.10.Kk
preprint: RIKEN-MP-08

## I Introduction

The study of quantum fields in Rindler space has played an important role in developing our understanding of quantum fields in non-trivial space-times. The importance of these studies derives to a large extent from the existence of a horizon in Rindler space. With its close connection to Minkowski space via the known local relation between fields in uniformly accelerated and inertial frames, Rindler space provides the simplest possible context to investigate kinematics and dynamics of quantum fields close to a horizon.

The kinematics of non-interacting quantum fields in Rindler space FULL73 (), BOUL75 (), DAVI75 () and their relation to fields in Minkowski space together with the interpretation in terms of quantum fields at finite temperature UNRU76 (), SCCD81 () are well understood. The relation between acceleration and finite temperature remains an important element of the thermodynamics of black holes. Although formally established, the relation between fields in Rindler and Minkowski spaces has posed some intriguing interpretational problems. Here we mention in particular the issue of compatibility of the radiation generated by a uniformly accelerated charge observed in Minkowski space and the apparent absence of radiation in the coaccelerated frame. This problem was finally resolved by the realization that the counterpart of Minkowski space radiation is the emission of zero energy photons in Rindler space HIMS921 (), HIMS922 (), REWE94 (). Dynamical issues of quantum fields have received less attention (cf. CRHM07 ()). Related to topics to be discussed in the following are the study of the relation between interacting quantum fields in Minkowski and Rindler spaces within the path-integral formalism UNRU84 (), the calculation of level shifts in accelerated hydrogen atoms PASS98 () and the investigations of the decay of protons MULL97 (), VAMA00 ().

Dynamical issues of quantum fields in Rindler space are in the center of our work. Subject are static forces and their properties relevant for the structure of matter in Rindler space. We will study the forces acting between static scalar sources, electric charges and massive sources generated by exchange of massless scalar particles, photons and gravitons respectively. The relevant quantities involved are the corresponding static Rindler space propagators. While the time dependent propagators in Rindler and Minkowski spaces are trivially related to each other by a coordinate transformation with corresponding mixing of the components, this is not the case for the static propagators obtained by integration over the corresponding time. The different properties of the static propagators reflect the significant differences of Rindler and Minkowski space Hamiltonians. Though it was realized from the beginning that the interpretation of Rindler and Minkowski particles (cf. e.g. FULL73 ()) is different, the differences in the corresponding Hamiltonians have not received sufficient attention. (The identity of the Rindler and Minkowski space Hamiltonians for the frequently used special case of a massless field in two dimensional space-time may have obscured the issue.) As shown in LOY08 (), the Rindler space Hamiltonian exhibits symmetries which give rise to a highly degenerate spectrum. In particular this degeneracy is also present in the zero energy sector. This 2-dimensional sector of zero modes is the image of the photons generated by the charges uniformly accelerated in Minkowski space. It will be shown to give rise to an unexpected quantum mechanical contribution to the force acting between two charges. Also the classical Coulomb-like contribution to the force will be shown to significantly deviate from the electrostatic force in Minkowski space. The electrostatic force will be determined via Wilson loops and Polyakov loop correlation functions. This method will enable us to separate the contribution of the quantum mechanical transverse photons from that of the classical longitudinal field. It will be the method of choice if one attempts to determine the static force in simulations of non-abelian gauge theories on a Rindler space lattice.

The modifications in the interaction of two charges at rest in Rindler space raise naturally the question of the stability of atomic systems in Rindler space. We shall investigate this issue for an ensemble of hydrogen atoms and show within a non-relativistic reduction that indeed only metastable states exist. We will estimate the probability of ionization as a function of the distance to the horizon and present arguments concerning the stability of other forms of matter.

## Ii Propagators and interactions of scalar particles in Rindler space

### ii.1 The quantized scalar field in Rindler space

The Rindler space metric RIND01 ()

 ds2=e2aξ(dτ2−dξ2)−dx2⊥, (1)

is the (Minkowski) metric seen by a uniformly accelerated observer (acceleration in -direction). Rindler space () and Minkowski space () coordinates are related by

 t(τ,ξ)=1aeaξsinhaτ,x1(τ,ξ)=1aeaξcoshaτ. (2)

The range of is

 −∞≤τ,ξ≤∞, (3)

while the preimage of the Rindler space covers only part of Minkowski space, the right “Rindler wedge”,

 R+={xμ∣∣|t|≤x1}. (4)

The restriction of the preimage of the Rindler space to the right Rindler wedge gives rise to a horizon, the boundary . With this property the Rindler metric can be identified with other static metrics in the near horizon limit. In particular this is the case for the Schwarzschild metric which can be approximated in the limit that the distance from the horizon is small in comparison to the Schwarzschild radius and if the spherical Schwarzschild horizon is replaced by a tangential plane.

We start our study of propagators and interactions with a discussion of a scalar field in Rindler space (cf. for instance FULL73 (), CRHM07 (). We will use the notation of LOY08 ()). The action of a free massive scalar field is given by

 S=12∫dτdξd2x⊥{∂τϕ)2−(∂ξϕ)2−(m2ϕ2+(∂⊥ϕ)2)e2aξ}. (5)

The wave equation

 [∂2τ−Δs+m2e2aξ]ϕ=0, (6)

with the Laplacian

 Δs=∂2ξ+e2aξ∂2⊥, (7)

is solved by the normal modes (in terms of the McDonald functions)

 ϕω,k⊥(ξ,x⊥)=1π√2ωasinhπωaKiωa(1a√m2+k2⊥eaξ)eik⊥x⊥, (8)

which form a complete, orthonormal set of functions. The normal mode expansion of the scalar field reads

 ϕ(τ,ξ,x⊥)=∫dω√2ωd2k⊥2π(a(ω,k⊥)ϕω,k⊥(ξ,x⊥)e−iωτ+a†(ω,k⊥)ϕ⋆ω,k⊥(ξ,x⊥)eiωτ) (9)

with the creation and annihilation operators . The stationary states associated with the Hamiltonian (

 HR = 12∫dξd2x⊥{π2+(∂ξϕ)2+e2aξ(m2ϕ2+(∂⊥ϕ)2)} (10) = ∫d2k⊥∫∞0dωωa†(ω,k⊥)a(ω,k⊥),

including the lowest energy () state, are degenerate with respect to the value of the transverse momentum . This degeneracy has its origin in the appearance of the transverse momentum (in combination with the mass) as a coupling constant of the “inertial potential” in the Hamiltonian.

In the Rindler wedge (4), the field operator can be represented in terms of plane waves in Minkowski space or in terms the Rindler space normal modes (9) resulting in the representation of the Rindler creation and annihilation operators in terms of the corresponding Minkowski space operators. This relation yields the important result for the expectation value of the Rindler number operator in the Minkowski ground state

 ⟨0M|a†(ω,k⊥)a(ω′,k′⊥)|0M⟩=1e2πωa−1δ(ω−ω′)δ(k⊥−k′⊥) (11)

which exhibits a thermal distribution with temperature

 T=1β=a2π. (12)

It is important to realize that the infinite degeneracy of the Rindler eigenstates leads, at long wavelengths, effectively to 1-dimensional thermal distributions with weight  .

### ii.2 The scalar Rindler space propagator

The propagators in Minkowski and Rindler spaces are related to each other by the coordinate transformation (2). Written in terms of Rindler and Minkowski coordinates respectively, the propagator of a massless field is given by (cf. TRDA77 (), DOWK78 ())

 D(τ−τ′,ξ,ξ′,x⊥−x′⊥)=i⟨0M∣∣T[ϕ(τ,ξ,x⊥)ϕ(τ′,ξ′,x′⊥)]∣∣0M⟩=D(x−x′) =14iπ2((x−x′)2−iδ)=a2e−a(ξ+ξ′)8iπ21cosha(τ−τ′)−coshη−iδ, (13)

with the notations

 D(τ−τ′,ξ,ξ′,x⊥−x′⊥)=D(x(τ,ξ,x⊥)−x(τ′,ξ′,x′⊥)) (14)

and

 coshη(ξ,ξ′,x⊥−x′⊥)=1+(eaξ−eaξ′)2+a2(x⊥−x′⊥)22ea(ξ+ξ′)=1+σ2(ξ,ξ′,x⊥−x′⊥). (15)

As we will see, it is (or equivalently ) which determines the interaction energies rather than the proper distance in Rindler space

 d(ξ,ξ′,x⊥−x′⊥)=√1a2(eaξ−eaξ′)2+(x⊥−x′⊥)2. (16)

The quantity actually can be viewed as the proper distance in the 4-dimensional AdS space. The appearance of this geometry (of a three-hyperboloid) has been noted SCCD81 () in the context of the unusual density of states in the photon energy momentum tensor in Rindler space. The quantity or equivalently exhibits the remarkable invariance under the transformation

 ξ,ξ′→ξ+ξ0,ξ′+ξ0,x⊥,x′⊥→eaξ0x⊥,eaξ0x′⊥. (17)

The coordinate transformation (17) together with a rescaling of the fields leaves, for , the Hamiltonian (10) invariant and gives rise to the degeneracy of the spectrum. The invariance under scale transformations can be generalized to massive particles LOY08 ().

In Rindler space coordinates the 2-point function (13) satisfies

 (∂2τ−∂2ξ−e2aξ∂2⊥)D(τ,ξ,ξ′,x⊥)=δ(τ)δ(ξ−ξ′)δ(x⊥). (18)

After a Wick rotation of the Rindler time,

 τ→τE=−iτ, (19)

the propagator is periodic in the imaginary time with periodicity (cf. Eq. (12))

 DE(τE,ξ,ξ′,x⊥)=a2e−a(ξ+ξ′)8iπ21cosaτE−coshη−iδ. (20)

In the context of the electromagnetic field we will compute observables in both the real and imaginary Rindler time formalisms. (For discussions of “Euclidean” Rindler space propagators, in particular of their topological interpretation cf. CHDU78 (), LINE95 (), SVZA08 ().)

Without reference to the Minkowski space propagator, Eq. (13) can be derived alternatively via the normal mode decomposition (9) in Rindler space. With the help of Eq. (11) the result

 D(τ,ξ,ξ′,x⊥)=D(R)(τ,ξ,ξ′,x⊥)+i∫∞0dΩΩ∫d2k⊥(2π)2ϕΩ,k⊥(ξ,x⊥)ϕΩ,k⊥(ξ′,0⊥)cosΩτe2πΩa−1

is obtained where the propagator defined with respect to the Rindler space vacuum is given by

 D(R)(τ,ξ,ξ′,x⊥)=i⟨0R∣∣T[ϕ(τ,ξ,x⊥)ϕ(0,ξ′,0⊥)]∣∣0R⟩ =i∫∞0dΩ2Ω∫d2k⊥(2π)2ϕΩ,k⊥(ξ,x⊥)ϕΩ,k⊥(ξ′,0⊥)e−iΩ|τ|=a2e−a(ξ+ξ′)η4iπ2sinhη1a2τ2−η2−iδ. (22)

(For an interpretation of the difference between the two propagators in terms of image charges located in the left Rindler wedge , cf. CARA76 ().) Obviously, the propagator depends parametrically on the acceleration . In order to formulate properly the relation between propagators in Minkowski and Rindler spaces one has to define the Rindler space propagator for a fixed value of the acceleration at finite temperature (given by )

 Da,β(τ,ξ,ξ′,x⊥)=i1tre−βHR(a)tr{e−βHR(a)T[ϕ(τ,ξ,x⊥)ϕ(0,ξ′,0⊥)]}. (23)

In terms of the propagator , the central result concerning the relation between Rindler and Minkowski spaces is the identity

 i⟨0M∣∣T[ϕ(τ,ξ,x⊥)ϕ(0,ξ′,0⊥)]∣∣0M⟩=Da,2πa(τ,ξ,ξ′,x⊥), (24)

i.e., the Rindler space propagator defined with respect to the Minkowski ground state coincides with the Rindler space finite temperature propagator with the value of the temperature determined by the acceleration (cf. Eq. (12)). This identity makes also manifest that a change in the acceleration does not correspond to a change in temperature of the accelerated system. The acceleration appears not only as temperature in the Boltzmann factor but also as a parameter in the Hamiltonian (10) of the accelerated system. We will encounter observables which make explicit this twofold role of the acceleration.

The basic quantity in the following investigation of the properties and consequences of interactions generated by exchange of scalar particles and photons is the static propagator

 ~D(ω,ξ,ξ′,x⊥)=∫∞−∞dτeiωτD(τ,ξ,ξ′,x⊥). (25)

For vanishing mass the -integration in (LABEL:Dur) can be carried out in closed form (cf. RG65 () and EMOT53 ())

 ~D(ω,ξ,ξ′,x⊥)=a4πea(ξ+ξ′)sinhη(ei|ω|ηa+2isin|ω|ηae2π|ω|a−1). (26)

Alternatively, this result can be obtained by a contour integration of Eq. (13) in the complex plane. The first term is due to the pole infinitesimally close to the real axis while the second, imaginary contribution is the result of the summation of the residues of the poles at . The result for the static propagator and its decomposition into the real non-thermal and the imaginary thermal contributions read

 ~D(0,ξ,ξ′,x⊥)=~D(R)(0,ξ,ξ′,x⊥)+i2πβηea(ξ+ξ′)sinhη∣∣β=2πa=a4πea(ξ+ξ′)1sinhη[1+iηπ]. (27)

The imaginary part of the static propagator arises since the propagator (LABEL:Dur) is defined with respect to the Minkowski rather than to the Rindler ground state.

### ii.3 The interaction energy of scalar sources

Given the propagator, the interaction energy between two scalar sources is obtained by adding to the action (5) the scalar particle-source vertex

 Sint=−∫dv0Hint=κ0∫d4v√|g(v)|ρ(v)ϕ(v), (28)

where stands for either Minkowski () or Rindler coordinates (). The effective action (the generating functional of connected diagrams) associated with the source is given by

 Wsc=−12κ20∫d4v√|g(v)|ρ(v)∫d4v′√|g(v′)|ρ(v′)D(v,v′). (29)

For two point like sources moving along the trajectories which are parametrized in terms of their proper times , we find

 √|g(v)|ρ(v)=∑i=1,2∫dsiδ4(v−vi(si)). (30)

We assume the sources to be at rest in Rindler space and evaluate by expressing the proper times by the coordinate times and obtain for sources positioned at

 Wsc=−12∑i,j=1,2κ(ξi)κ(ξj)∫dτi∫dτ′jD(τi−τ′j,ξi,ξj,xi⊥−xj⊥). (31)

We have introduced the effective coupling constant

 κ(ξ)=eaξκ0, (32)

which, due to the difference between proper and coordinate times,“runs” with the coordinates of the charges. With the sources at rest, depends only on the differences of the times . After carrying out the integrations, up to a factor , the size of the interval in the integration over the sum of the times, is determined by the static propagator and yields for the interaction energy of two scalar sources

 Vsc = −κ(ξ1)κ(ξ2)~D(ξ1,ξ2,x⊥1−x⊥2)=−aκ204πsinhη[1+iηπ], (33)

where (cf. Eq. (15)). For two sources at rest in Minkowski space, this procedure yields the interaction energy .

The interaction energy (33) constitutes an explicit example of the bivalent role of the acceleration . The dependence of the real part of the interaction is exclusively due to the dependence of the Hamiltonian (10) on the parameter while the imaginary part depends in addition on the acceleration via the temperature (cf. Eq. (27)). The appearance of a non-trivial imaginary contribution to the “static interaction” generated by exchange of scalar particles and, as we will see also by photons or gravitons, is a novel phenomenon not encountered in the static interactions in Minkowski space. Here we will analyze this phenomenon. Other properties of the interaction (33) will be discussed later in the comparison with the “electrostatic” interaction.

As follows from Eq. (18) the static propagator (27) satisfies the Poisson equation for a point-like source in Rindler space. Since the source is real, the imaginary part of the propagator satisfies the corresponding (homogeneous) Laplace equation. In turn, this implies that the imaginary part of propagator or the interaction energy can be represented by a linear superposition of zero modes of the Laplace operator (7). From Eq. (LABEL:Dur) we read off

 Im~D(0,ξ,ξ′,x⊥−x′⊥) = aη4π2ea(ξ+ξ′)sinhη (34) = 14π3a∫d2k⊥eik⊥(x⊥−x′⊥)K0(k⊥aeaξ)K0(k⊥aeaξ′).

It is instructive to compare the Rindler space propagator with the finite temperature propagator in Minkowski space. As above we decompose the propagator of a non interacting scalar field into thermal and non-thermal contributions

 Dβ(x) = i1tre−βHMtr{e−βHMT[ϕ(x)ϕ(x′)]} (35) = im4π2√−x2+iϵK1(m√−x2+iϵ)+δDβ(x),

carry out the Fourier transform of the thermal part

 δ~Dβ(ω,x)=∫∞−∞dtδDβ(x)eiωt=i2πxθ(ω2−m2)sin(√ω2−m2x)eβ|ω|−1, (36)

and obtain for massless particles

 m=0,~Dβ(0,x−x′)=∫∞−∞dtDβ(t,x−x′)=14π|x−x′|+i2πβ. (37)

The Fourier transformed thermal propagators in Rindler (27) and in Minkowski space (37) become identical to order apart from the factor and differ from the corresponding ground state contribution only by the constant . The convergence to this limit is not uniform in Rindler space, since it requires . Due to the twofold role of the acceleration the finite temperature contribution to the propagator in Minkowski space is only part of the leading order correction to the propagator (cf. Eq. (27)) in Rindler space. The structure of the Minkowski space static propagator suggests that the imaginary part is due to on-shell propagation of zero-energy massless particles. The difference between Rindler and Minkowski space propagators is due to the different dimensions (0 and 2 respectively) of the space of zero modes. Furthermore, while the Minkowski space zero mode is constant, the zero modes in Rindler space exhibit a non-trivial dependence on all the three coordinates. Finally for massive particles no zero mode exists in Minkowski space (Eq. (36)) while in Rindler space together with the degeneracy in the spectrum also the zero modes persist. In this case the spectral representation in Eq. (34) remains valid provided we replace in the arguments of the McDonald functions.

The imaginary part of the propagator determines the particle creation and annihilation rates. To leading order in the coupling constant , the probability for a change in the initial state in the time interval is given by (cf. Eqs. ((28)-(32))

 ≈Im∫ττ0dτ′∫ττ0dτ′′∑i,j=1,2κ(ξi)κ(ξj)D(τ′−τ′′,ξi,ξj,x⊥i−x⊥j). (38)

The rate for a change in the initial state within an arbitrarily large time interval is easily obtained to be

 1TPex(T,0) → Im∑i,j=1,2κ(ξi)κ(ξj)~D(0,ξi,ξj,x⊥i−x⊥j) (39) = κ20a2π2(1+η(ξ1,ξ2,x⊥1−x⊥2)sinhη(ξ1,ξ2,x⊥1−x⊥2)).

The total response (not observable in the accelerated frame) of the field to external sources determines the imaginary part of the static propagator. Furthermore by defining the reaction rate in terms of the proper time of the external sources (cf. HIMS921 (), HIMS922 (), CRHM07 (), REWE94 ()) it is seen that the imaginary part is determined by the total rate for Bremsstrahlung of uniformly accelerated sources observable in Minkowski space.

These results imply, that under the transformation from the inertial to the accelerated system comoving with the uniformly accelerated charge, the particles generated in Minkowski space by Bremsstrahlung are mapped into zero energy excitations in Rindler space. In this way the conflict between particle production in Minkowski space and the conservation of energy in Rindler space in the presence of static sources is resolved. The on-shell zero modes describe the radiation field in Rindler space without any change in the energy in Rindler space and may be viewed as a “polarization” cloud of on-shell particles induced by external sources or by classical detectors (cf. GROV86 (), MAPB93 (), UNRU92 (), HURA00 ()). Similar remarks apply for the acceleration induced decay of protons MULL97 (), VAMA00 (). Essential for this important role of the zero modes is, as indicated above, the peculiar symmetry of the Rindler Hamiltonian which gives rise to the extensive degeneracy.

## Iii Wilson and Polyakov loops of the Maxwell field

### iii.1 Wilson loops in Minkowski and Rindler space

In this section we consider the Maxwell field coupled to external charges given by the action

 S=−14∫dτdξd2x⊥√|g|FμνFμν+Sint,Sint=∫dτdξd2x⊥√|g|Aμjμ. (40)

With changing emphasis we will describe various methods for evaluating the interaction energy of static sources. The computation of Wilson loops BAMU94 () and Polyakov loop correlation functions JASM02 () constitute the preferred techniques in analytical and numerical studies of interaction energies of static sources in gauge theories. In the comparison of these two methods the emphasis will be on the consequences of the Wick rotation to imaginary time for the interaction energy which will be seen to be of relevance also for static interactions in Yang-Mills theories. Evaluation of the Wilson loops in different gauges will enable us to identify the origin of real and imaginary parts respectively of the electrostatic interaction.

Wilson loops are defined as integrals over the gauge field along a closed curve in space-time

 eiWC=eie∮CdxμAμ. (41)

The invariance of the Wilson loop under gauge and (general) coordinate transformations and reparameterization which is explicit in Eq. (41) makes the Wilson loop a particularly useful tool for our purpose. Up to self energy contributions, the interaction energy of two oppositely charged sources is given by the expectation value (e.g. in the Minkowski space ground state) of a rectangular Wilson loop in a time-space plane with side lengths and

 Σ±=limT→∞1T˜WC[R,T], (42)

with the ground state expectation value

 e˜WC[R,T]=⟨0M|eiWC[R,T]|0M⟩. (43)

The gauge fields along the loop can be interpreted as resulting from two opposite charges which are separated in an initial phase from distance to (for a rectangular loop this initial phase is reduced to one point in time), remain separated at this distance for the time and recombine in a final phase. In order to make the contributions from the turning-on period negligible, the interaction energy of static charges is defined by the limit. In terms of the photon propagator

 Dμν(x,x′)=i⟨0M∣∣T[Aμ(x)Aν(x′)]|0M⟩, (44)

the Wilson loop is given by (cf. BAMU94 ())

 ˜WC=12e2∫ds∫ds′dxμCdsdx′νCds′Dμν(xC(s),x′C(s′)). (45)

In Lorenz gauge,

 ∂μAμ=0,

the Minkowski space photon propagator is expressed in terms of the scalar propagator (cf. Eq. (13)) as

 DMμν(x,x′)=ημνD(x,x′)=ημν4iπ2[(x−x′)2−iδ], (46)

with the Minkowski space metric . The Rindler space photon propagator is obtained by the change in coordinates (2)

 Dμν(τ−τ′,ξ,ξ′,x⊥−x′⊥) = λμν(v,v′)D(x(v),x′(v′)) (47) = a2e−a(ξ+ξ′)8iπ2λμν(v,v′)cosha(τ−τ′)−coshη−iδ,

where we have used the notation

 v(′)={τ(′),ξ(′),x(′)⊥},λμν(v,v′)=dxρdvμdx′σdv′νηρσ. (48)

Under the coordinate transformation, the Lorenz gauge condition becomes

 ∇μAμ=∂τAτ+(∂ξ+2a)Aξ+∂⊥A⊥=0,

with the covariant derivative .

### iii.2 Interaction energy of static charges in Rindler space

#### iii.2.1 Wilson loops of gauge fields in Lorenz gauge

Invariance of the Wilson loop under coordinate transformations does not imply invariance of the interaction energy. Under the coordinate transformation (2) the shape of a loop changes, as is illustrated in Fig. 1 for the case of a rectangular loop in Rindler space. With the change in shape also the value of the interaction energy changes which is defined with respect to two different limits ( or ).

We compute the interaction energy for a rectangular loop with 2 of the 4 segments of the loop varying in time and kept fixed while the other two segments are computed at fixed . The sum of the two contributions from the integration along the axis can be carried out without specifying the segments in the spatial coordinates. Inserting (cf. Eq. (47))

 D00(τ−τ′,ξ,ξ′,x⊥−x′⊥)=a28iπ2cosha(τ−τ′)cosha(τ−τ′)−coshη−iδ (49)

into Eq.  (45) we find

 W0=e2a28iπ2∫T0ds∫T0ds′[cosha(s−s′)cosha(s−s′)−(1+iδ)−cosha(s−s′)cosha(s−s′)−(coshη+iδ)]. (50)

Here is given by (15) with the spatial coordinates of the vertices of the rectangle. Introducing as integration variables can be rewritten as

 W0=e2a28iπ2∫T0ds[I0(s,1+iδ)−I0(s,coshη+iδ)], (51)

with

 I0(s,coshη)=2∫s0ds′coshas′coshas′−coshη−iδ=2s+2iπcoshηa√sinh2η+2iδ ⋅⎡⎢ ⎢⎣1−iπ⎛⎜ ⎜⎝lneas−coshη−√sinh2η+2iδeas−coshη+√sinh2η+2iδ+ln1−coshη+√sinh2η+2iδcoshη+√sinh2η+2iδ−1⎞⎟ ⎟⎠⎤⎥ ⎥⎦. (52)

The last step can be verified by differentiation. The contribution to the interaction energy of two oppositely charged sources can be extracted in the limit from the two  - independent terms in (52) (the integrals of the - dependent terms converge).

For large , the integration along the spatial segments of the rectangle (the horizontal segments of the loop in Fig. 1) yields a -independent term and does therefore not contribute to the interaction energy. The non-diagonal element gives rise to two space-time contributions to the Wilson loop which in the large limit become independent of the spatial coordinates and cancel each other. Thus we obtain the asymptotic value of the Wilson loop expressed in terms (15)

 Σ±(σ)=limT→∞1TW0=−e24πacothη[1+iηπ]+U0=V(σ)+U0, (53)

with the interaction energy

 V(σ)=−e24πa1+σ2√2σ2+σ4[1+iπln√2σ2+σ4+σ2√2σ2+σ4−σ2]. (54)

The integration “constant” arises from the first term in (50) and represents the self-energy of the static charges. Regularizing the divergent integrals by point splitting, is given in terms of the proper distance in AdS (cf. Eq. (15))

 U0=e2a8π(1√2δσ(ξ)+1√2δσ(ξ′)+2iπ),δσ2(ξ(′))=a22(δξ2+e−2aξ(′)δx2⊥). (55)

Before discussing the properties of the “electrostatic” interaction and the comparison with the static scalar interaction we briefly describe a simple alternative method for calculating the interaction energy of two static charges. In analogy with the corresponding calculation for scalar fields (cf. Eqs. (28)-(32)) the current in Eq. (40) generated by two charges moving along the trajectories is parametrized as

 √|g|jμ(v)=∑iei∫dsidvμi(si)dsiδ4(v−vi(si)), (56)

resulting in the photon-charge vertex

 Sint=∑iei∫dsiAμ(vi(si))dvμidsi. (57)

As for the scalar case (Eqs. (28)-(32)), the relevant quantity to be computed is the effective action which, for charges at rest in Rindler space, yields the sum of interaction and self energies

 Wvc = 12∑i=1,2eiej∫dsi∫dsjDμν(vi(si),vj(sj))dvμi(si)dsidvνj(sj)dsj (58) = 12∑i,j=1,2eiej∫dτi∫dτjD00(τi−τj,ξi,ξj,xi⊥−xj⊥),

where the propagators in different coordinates are obtained from each other by the corresponding coordinate transformations (cf. Eq. (47)). Unlike in the scalar case, for exchange of photons no factor renormalizes the coupling constants when changing from the proper to coordinate time . We define the Fourier transform in time of the -component of the propagator (Eq. (49)) by the limit

 ~D00(ω,ξ,ξ′,x⊥) = limT→∞∫T−TdτeiωτD00(τ,ξ,ξ′,x⊥) (59) = a2sinωT4iπ2ω+a4πcothη[e−iωηa+2isinωηa1−e−2πωa],

and disregarding the (divergent) constant, for opposite charges , the result (54)

 −e2~D00(0,ξ,ξ′,x⊥)=V(σ), (60)

is reproduced.

In Fig. 2 are plotted real and imaginary parts of the interaction energy of two static charges in comparison with the interaction energy of two scalar sources. The distance which determines both scalar (33) and vector (54) interaction energies is the proper distance of the AdS space and not of Rindler space, i.e., the static interaction energies generated by scalar particle and photon exchange are invariant under the scale transformation (17). At small distances where the effect of the inertial force is negligible both interaction energies display the familiar linear divergence in the real part and as in Minkowski space, vector and scalar exchange give rise to the same behavior. For small distances the imaginary part is constant and agrees with the constant in the finite temperature propagator (37) in Minkowski space

 limσ→0V(σ) = (61) limσ→0Vsc(σ) = (62)

With increasing distance the inertial forces become important. They weaken both scalar and vector interaction energies and change completely the asymptotic behavior

 limσ→∞V(σ) = −e2a4π(1+12σ4+iπln2σ2), (63) limσ→∞Vsc(σ) = −κ20a4π1σ2(1+iπln2σ2). (64)

Significant differences in the asymptotics for scalar and vector exchange are obtained. Qualitative differences are observed for the imaginary part which vanishes asymptotically for scalar exchange and diverges logarithmically with for photon exchange. Given the highly relativistic motion of the sources in Minkowski space, spin effects are expected to be important. Indeed the comparison of Eqs. (33) and (53) shows that the differences between scalar and vector exchange are due to the additional factor in (53) arising from the transformation of the propagator from Minkowski to Rindler space. In this context we also note the stronger asymptotic suppression of the “electrostatic” force in comparison to the force generated by scalar exchange. From the point of view of a Minkowski space observer (cf. Eq. (58)), cancellation between magnetic and electric forces generated by the spatial components of the current and by the charges respectively is to be expected.

Of particular interest is the behavior of the interaction energy if one of the sources approaches the horizon while the position of the other source is kept fixed. For

 ξh→−∞,σ2→12e−a(ξh+ξ)(e2aξ+a2(x⊥−x′⊥)2)→∞, (65)

the interaction energies are given by

 limξh→−∞V(σ) = −e2a4π(1−ia(ξh+ξ)π), (66) limξh→−∞Vsc(σ) = −κ20a2πea(ξh+ξ)1e2aξ+a2(x⊥−x′⊥)2(1−ia(ξh+ξ)π). (67)

In both cases the interaction is dominated by the imaginary radiative contribution and no bound states are expected in this regime. We also find in analogy with the “no hair” theorem BEKE72 (), BEKE98 (), TEIT72 () for Schwarzschild black holes that a scalar source close to the horizon cannot be observed asymptotically while vector sources are visible. Formally the difference arises from the running of the scalar coupling (32) to zero when approaching the horizon while the electromagnetic coupling remains constant. In detail the results for Rindler and Schwarzschild metrics are different. A significant improvement can be obtained by modifying the Rindler metric

 ds2=e2aξ(dτ2−dξ2)−dx2⊥→e2aξ(dτ2−dξ2)−14a2dΩ2. (68)

The dynamics in this space is very close to the dynamics in Rindler space unless the difference between the and matters. This is the case when deriving the “no hair” theorem where only the waves matter COWA71 ().

#### iii.2.2 Wilson loops of gauge fields in Weyl gauge

In the (covariant) Lorenz gauge only the -component of the propagator contributes (cf. Eqs. (54) and (60)) which simplifies significantly the evaluation of the electrostatic interaction energy and by the same token hides the difference in origin of its real and imaginary parts. Separation of constrained and dynamical variables becomes manifest in Weyl gauge

 A0=0.

In this gauge one works with two unconstrained dynamical fields describing the photons and a longitudinal field constraint by the Gauß law. The gauge field is decomposed accordingly

 Ai=∂iχ+^Ai, (69)

into the longitudinal component given by the scalar field and the transverse field satisfying

 Pij^Aj=^Ai,withPij=δij+∂i1Δ∂j, (70)

and the Laplacian

 Δ=−∂i∂i=e−2aξ(∂2ξ−2a∂ξ)+∂2⊥. (71)

Longitudinal and transverse fields are not coupled to each other. Their actions are given by


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# Could explosives replace drilling for science and construction in space?

Drilling seems to require heavy equipment, preferably with humans on site for maintenance, much energy and might be even more problematic in low gravity and unknown geology. Deepest hole drilled in space was 292 cm by Apollo 17, manually operated on site.

Could a heavy drill be replaced by a rover dropping off a much lighter shaped charge, taking cover when it explodes, and returning to examine the debris from the deep which landed on the surface? Couldn't a modern shaped charge even cause a deep narrowish hole, rather than a flat crater, which could be inspected by "a ChemCam" and even by physically lowering down small instruments and tools? And then even by lowering down another explosive, much like construction and mining drilling on Earth often is done only in order to place explosives more favourably.

By blasting out a bay shaped cavity in a Martian or Lunar hill, radiation shielding and maybe thermal benefits might be provided for a habitat module later placed there. Explosives might make subsurface water ice much more available to a human space colony. With the advantage of using only a few tons of explosives which require almost no operational overhead, other than being placed by a rover. Even if a detonation gives an unwanted result, the investment lost was low compared to a heavy drilling station, and could be replaced by a new attempt on another site nearby.

What are the main problems which I optimistically leave out here?

Actual examples of explosives in space: Hayabusa2 will use a shaped explosive charge to create a crater from where it will land and take a sample (animation and an abstract of a paywalled article). That is more of a replacement for an impactor like Deep Impact, rather than for a drill. Except for explosive bolts, the only other use of (non-military) explosives in space I know of is Apollo 14, 16, 17 and that was for seismic purposes.

EDIT: Is there a way to roughly compare the launch mass and useful energy delivered of explosive devices compared to mining devices on the Moon, Mars, Asteroids given some performance criteria such as depth reachable or mass of rocks cleared? I imagine that explosives are pretty energy intensive payloads compared to solar powered machines.

• I believe you have the right idea burrowing in a lunar hill. But I believe Greg Baiden's robots are the way to accomplish this. See youtube.com/watch?v=FVmD3s1DODI – HopDavid Dec 27 '14 at 18:25
• @HopDavid Baiden talks there about such requirements as: delineation drill, development drill, some form of digging machine, explosives loading unit and a sintering machine. So quite a bit of heavy machinery and that's without the transport of excavated (dug, drilled or blasted) materials to the surface. He then goes on to describe machinery needed: digging machine, dump truck, spray-on machine, and a combined drilling and explosives unit. Am I missing something here? – TildalWave Dec 27 '14 at 18:39
• Baiden believes he can size the equipment to fit within launch mass constraints. But yes, it would be an ambitious undertaking. – HopDavid Dec 28 '14 at 14:31
• @HopDavid I personally don't think that would work very well. Mars, the Moon, asteroids et al. are rather cold places beneath the surface and there's only so much of miniaturization that can be done before brittleness of thin materials in extremely cold environments becomes an unworkable problem. It would also scale operations down along with it and with some things you can't construct in a modular fashion or ask whoever ordered the parts to also send smaller astronauts up there that will use smaller habitats because you can't build larger ones as the tools you built don't reach the ceiling :) – TildalWave Dec 30 '14 at 0:45

There's one quite peculiar application where explosives will not require much post-processing - using a nuclear charge to create a cavern under ground. While it may be considered illegal in terms of international treaties, it is a viable construction technique. Melt cavity radius would be roughly proportional to the cube root of the device's yield :

$r_{\textrm{(in meters)}} \approx 16.76 Y_{\textrm{(in kilotons)}}^{\frac{1}{3}}$ for some of Earth's rock types

However, a fully contained cavity at the current technology level requires drilling and concrete-filling as pre-processing steps. It might be possible to devise a ballistic/rocket-assisted penetrator contraption to dig into the ground from a hyperbolic orbit, but there's always the chance (4 per cent for the Earth, higher for less-studied planetary bodies) that the hole may be unsuitable, and another risk is penetration failure.

To sum it up: at the current level of technology nuclear explosions will likely require drilling. Future advances may replace drilling with ballistic penetrators.

For the record, I am not insensitive to the environmental and biological risks posed by nuclear explosions. An unsuccessful launch abort may set up the stage for another Broken Arrow. The explosion may poison aquifers (if they exist) or vent contaminants into the atmosphere, or kill local life. The real reason I added this answer is we have to always keep in mind the whole gamut of techniques the humanity possesses. Nukes pack enormously large energy in low volume and mass.

For scientific purposes, like exposing near-surface materials, probing with seismic explosives, or prospecting for pockets or veins of subsurface materials, I guess it could work, and you already give a few good examples in your question. But for industrial purposes, on its own and without rolling in heavy machinery once you, say, reduce some heavy boulder into rubble, not very likely. Precision required is too great and there's too many uncertainties involved, some, you'd only make worse with the use of explosives;

Explosives are otherwise used all the time in both subsurface and surface blast-mining on Earth. Quarries would also be one such example. But laying and precise detonating charges in sequence so they also clear the resulting craters off ruble would be incredibly difficult and it wouldn't result in perfectly clear tunnels to roll into them some form of autonomous or remotely operated vehicle to place new charges either. No rock is exactly homogenous and fault-free, and with each explosion you're deepening these cracks that can give-in at any time. So you'd be laying support structures (roofing and walls support) that requires heavy machinery, after you managed to clear the path for them, usually using smaller, but still heavy loaders and trucks or carts (which would also require rails, again, laid by heavy machinery). The deeper you go, the tougher this gets and you're constantly laying your charges along its way, increasing stress with each detonation. But, I guess, you could start like this, then roll in heavy machinery as you need to deepen the hole.

So, in a nutshell, not really. At least not for construction work or heavy mining operations. But you could manage to crack open tougher regolith to expose subsurface pockets of water ice, use smaller amounts of it as seismology charges, explosive bolts, and so on.

• Even "taking a bite" out of, as opposed to caving into, a hillside leaving no roof and with slanting sides, even that would protect against most of the cosmic radiation and solar heating from three sides. Maybe the rubble left could be largely cleared by a new set of explosives applied to blast it out in the open? – LocalFluff Dec 27 '14 at 19:32
• @LocalFluff Yes, that would probably work, especially in smaller gravity environment and on frozen ground. Or the rubble could be compressed and flattened to provide a more sturdy surface to build on. But the cliffs would still have to be secured somehow to prevent later avalanche of debris falling on top of you. Note that here lower average surface temperatures work against you and the rock is more brittle due to it. And the large temperature difference between the parts exposed to the Sun and the ones in shadow could further destabilize it due to thermal expansion. – TildalWave Dec 27 '14 at 19:39
• BTW, re your edit, it would probably be easier making explosives in-situ than transporting them between celestials (maybe a good topic for a new question?). And re blasting an alcove, to prevent as much of sherd as possible I'd suggest blasting through already existing fault-lines of suitable geometry. On Mars, that might mean simply removing excess aeolian sedimentation (read: compacted sand) on the sides of steep slopes or cliffs. – TildalWave Dec 27 '14 at 20:05

It depends on what you are trying to achieve.

Depending on the condition of what is being drilled, drilling a hole will produce a neat hole of uniform diameter. Explosives however, always produce a significant zone of fractured material around where the explosive was detonated.

If you want to take a sample of anything, drilling is the best option.

If you want to produce holes for foundations for stilts, drilling is still the best option.

If you want to maximize a crater it is still better to drill a hole, or holes, to the required depth, fill the holes with explosives, as required and then blast.

By burying an explosive in a drilled hole all the energy of the blast is direct to fracturing the rock and moving the rock. By detonating an explosive on the surface a significant portion of the energy of the explosive will be lost, even in a vacuum, which is inefficient.

On Earth we tend to be wasteful because it is easy to obtain resources. In Space however, resources are scarce and their efficiency needs to be maximized.

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Fluency exercise

## Solution

A square is enclosed by the lines:

1. $\mathbf{r}=\begin{pmatrix}0\\2\end{pmatrix} + a\begin{pmatrix}1\\1\end{pmatrix}$
1. $\mathbf{r}=\begin{pmatrix}0\\6\end{pmatrix} + b\begin{pmatrix}-1\\1\end{pmatrix}$
1. $\mathbf{r}=\begin{pmatrix}2\\1\end{pmatrix} + c\begin{pmatrix}2\\2\end{pmatrix}$
1. $\mathbf{r}=\begin{pmatrix}1\\2\end{pmatrix} + d\begin{pmatrix}1\\-1\end{pmatrix}$

Can you convince yourself that the adjacent edges are perpendicular?

We need to consider the direction vector of each line. Lines (1) and (2) do not have the same direction vector and so they will cross at some point. If we now think about the behaviour of each line as it moves away from the crossing point we can show that they are perpendicular using the diagram below.

Can you use a similar argument to convince yourself about the other adjacent edges of the square?

Find the area of the square.

To do this we need to identify the positions of some of the vertices. We could aim to find the length of one side by finding two adjacent vertices, or we could find the length of the diagonal by identifying vertices opposite each other. It doesn’t matter which pair we choose to find so let’s say we look for vertices A and B shown on the diagram below.

To find the position of point A, we need to find the point that is common to equations (1) and (4), (they need to have the same position vector $\mathbf{r}$). With that in mind, we can write down that $a=1+d$ and $2+a=2-d$. Solving these equations simultaneously we find that $a=\frac{1}{2}$ and $d=-\frac{1}{2}$.

Substituting these values into either equation (1) or equation (4) we find the position vector of point A to be $\mathbf{r}=\begin{pmatrix}\frac{1}{2}\\\frac{5}{2}\end{pmatrix}$.

Repeating the process to find the position of point B, from equations (1) and (2) we can write down that $a=-b$ and $2 + a = 6 + b$. Solving these equations simultaneously we find that $a=2$ and $b=-2$.

Substituting these values into either equation (1) or equation (2) we find the position vector of point B to be $\mathbf{r}=\begin{pmatrix}2\\4\end{pmatrix}$.

We might have found the position of vertex B in another way. The line represented by vector equation (1) could have been constructed by starting at the position vector $\begin{pmatrix}0\\2\end{pmatrix}$ and ‘stepping out’ the direction vector as shown below. If the same process was then repeated to construct the line represented by vector equation (2) then we notice that the two lines intersect at point B with position vector $\begin{pmatrix}2\\4\end{pmatrix}$.

If the process is repeated to construct the lines represented by vector equations (3) and (4) then we notice that this pair intersect at $\begin{pmatrix}2\\1\end{pmatrix}$, forming the vertex opposite B in the enclosed square. These two vertices are aligned vertically and the line joining them forms the diagonal, length 3, of the enclosed square.

Now that we have the position vectors of vertices A and B we could find the length of the line AB using Pythagoras.

However, you might notice that the area of the square will be given by $AB^2$ and we can write this down directly, $AB^2 = 2 \times \left(\frac{3}{2}\right)^2 = \frac{9}{2}.$

You could also have noticed that the right-angled triangle joining points A and B is isosceles. This tells us that the square could be divided up into four of these triangles.

We can now show that the area of the square is again equal to $4 \times \frac{1}{2}\left(\frac{3}{2}\right)^2 = \frac{9}{2}.$

What can we now say about the positions of the other vertices and the diagonals of the square?

Find a vector equation for each of the two diagonals.

In order to write down a vector equation of a line we need to know the position of a point on that line and a direction vector. For the vertical diagonal of the square we know that point B, with position vector $\begin{pmatrix}2\\4\end{pmatrix}$ lies on the line and that the direction vector is $\begin{pmatrix}0\\1\end{pmatrix}$.

A vector equation of the vertical diagonal of the square is therefore $\mathbf{r}=\begin{pmatrix}2\\4\end{pmatrix} + p\begin{pmatrix}0\\1\end{pmatrix}.$

We could also have written $\mathbf{r}=\begin{pmatrix}2\\1\end{pmatrix} + p\begin{pmatrix}0\\1\end{pmatrix},$ or any vector equation of the form $\mathbf{r}=\begin{pmatrix}2\\s\end{pmatrix} + p\begin{pmatrix}0\\q\end{pmatrix}$ for any non-zero $q$.

The horizontal diagonal could be written as any vector equation of the form $\mathbf{r}=\begin{pmatrix}s\\\frac{5}{2}\end{pmatrix} + p\begin{pmatrix}q\\0\end{pmatrix}$ for any non-zero $q$.


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# A 63 kg kg person starts traveling from rest down a waterslide 6.0 mm above the ground. At the bottom of the waterslide, it then curves upwards by 1.0 mm above the ground such that the person is consequently launched into the air. Ignoring friction, how fast is the person moving upon leaving the waterslide

###### Question:

A 63 kg kg person starts traveling from rest down a waterslide 6.0 mm above the ground. At the bottom of the waterslide, it then curves upwards by 1.0 mm above the ground such that the person is consequently launched into the air. Ignoring friction, how fast is the person moving upon leaving the waterslide

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text:
# Rocket Thrust Instrumentation

### Help Support The Rocketry Forum:

#### JackO

##### MIT AeroAstro Undergrad -- HSRPL Founder
[FONT=&quot]I am recording the thrust of a large solid rocket engine with this load cell: [/FONT]https://www.robotshop.com/en/1000kg-butt ... -cell.html

[FONT=&quot]I believe I will need this amplifier: [/FONT]https://www.sparkfun.com/products/13879

[FONT=&quot]and this micro-controller: [/FONT]https://www.sparkfun.com/products/13261

[FONT=&quot]In order to record the thrust being administered to the cell over the course of a few seconds. Are these products compatible and appropriate? [/FONT]
[FONT=&quot]What other products will I need? What free software can I get to graph this data?[/FONT]

[FONT=&quot]Thanks all,[/FONT]

[FONT=&quot]Jack[/FONT]

#### AlphaHybrids

##### Well-Known Member
If you can gain access to the Research Forum there are threads there that describe what you are wanting to do.

1000kg? What size motor are you firing?

Edward

#### JackO

##### MIT AeroAstro Undergrad -- HSRPL Founder
As a minor, I cannot join the research forum. Since this topic is not on experimental propulsion but instrumentation, I was hoping I could get the information from this forum. We are first testing an M engine and later a P.

If you can gain access to the Research Forum there are threads there that describe what you are wanting to do.

1000kg? What size motor are you firing?

Edward

#### AlphaHybrids

##### Well-Known Member
That is a little load cell for that size motor, even if it is rated at 1000kg. The one I use for that size motors is 6-inches in diameter and 10,000 lb rating on it.

Edward

#### ttabbal

##### Well-Known Member
As a minor, I cannot join the forum, but since this topic is not on experimental propulsion but instrumentation, I was hoping I could get the information from this forum. We are first testing an M engine and later a P.

Your mentor doesn't have a test stand? Or is that part of the goal you have in mind?

The OpenScale will get the data for you, it has the HX711 onboard, so you don't need an extra one. You will need a computer or an OpenLog to log the data from the load cell if you use the OpenScale.

Note that if you are willing to write some code, an HX711 and an arduino board can be had on ebay much cheaper.

As for plotting the data, a spreadsheet can do it. Here's one of mine I've been experimenting with. Mine is only pressure right now, but something similar would work for thrust.

OpenOffice can do it, but it's slow at it if you have a lot of data. My ADC is much faster, 4Khz vs 80hz, so there are a lot more data points. https://www.analog.com/en/design-ce...tware/evaluation-boards-kits/EVAL-AD7190.html

#### JackO

##### MIT AeroAstro Undergrad -- HSRPL Founder
Alpha, unfortunately our budget would not allow for more than this bargain cell. If the P engine ends up making too much thrust in simulations; we will purchase two more and have the thrust push down on all three cells evenly.
That is a little load cell for that size motor, even if it is rated at 1000kg. The one I use for that size motors is 6-inches in diameter and 10,000 lb rating on it.

Edward

#### JackO

##### MIT AeroAstro Undergrad -- HSRPL Founder
Our mentor deals with liquid engines and is very busy with various contracts; he does not have a test stand that would fit our specific needs.
I'm willing to cough up the $30 for a pre-coded setup. Thanks very much for the links! To confirm, the Open Scale and load cell will be all that we will need to record the thrust? Your mentor doesn't have a test stand? Or is that part of the goal you have in mind? The OpenScale will get the data for you, it has the HX711 onboard, so you don't need an extra one. You will need a computer or an OpenLog to log the data from the load cell if you use the OpenScale. Note that if you are willing to write some code, an HX711 and an arduino board can be had on ebay much cheaper. As for plotting the data, a spreadsheet can do it. Here's one of mine I've been experimenting with. Mine is only pressure right now, but something similar would work for thrust. https://docs.google.com/spreadsheets/d/1vz2-BbZPnDetbAeja1M68vCV-hLnd3dhpZ1Yy2qc6iE/edit?usp=sharing OpenOffice can do it, but it's slow at it if you have a lot of data. My ADC is much faster, 4Khz vs 80hz, so there are a lot more data points. https://www.analog.com/en/design-ce...tware/evaluation-boards-kits/EVAL-AD7190.html #### ttabbal ##### Well-Known Member Our mentor deals with liquid engines and is very busy with various contracts; he does not have a test stand that would fit our specific needs. I'm willing to cough up the$30 for a pre-coded setup. Thanks very much for the links!
To confirm, the Open Scale and load cell will be all that we will need to record the thrust?

Those will get the load cell data into a serial data stream. To record it, you need either a computer and some code, or something like the OpenLog, which SF also sells. I believe Openscale includes the needed conversion to convert into pounds/kg as well.

I'm not sure about combining load cells for more capacity... And it seems like it would be cheaper to buy one big one. You are aware that a P motor will use a LOT of chemicals, right? That's big money even if you are making your own.

#### r66astro

##### Well-Known Member
TRF Supporter
I would use an amplifier that puts out a dc voltage then record with a Dataq they are only 39 bucks I think and usb to your laptop or computer, to record with.

#### JackO

##### MIT AeroAstro Undergrad -- HSRPL Founder
Those will get the load cell data into a serial data stream. To record it, you need either a computer and some code, or something like the OpenLog, which SF also sells. I believe Openscale includes the needed conversion to convert into pounds/kg as well.

I'm not sure about combining load cells for more capacity... And it seems like it would be cheaper to buy one big one. You are aware that a P motor will use a LOT of chemicals, right? That's big money even if you are making your own.
Ok great, thanks for the info.

I'm not sure of the multiple cell idea either, it came from one of our mentors, regardless one cell is sufficient for our M-engines.

We are; we will be searching for additional sponsors and utilising grants and fundraising programs after we characterize our fuel. Also, many companies offer student discounts which helps a lot.

-Jack

#### JackO

##### MIT AeroAstro Undergrad -- HSRPL Founder
I would use an amplifier that puts out a dc voltage then record with a Dataq they are only 39 bucks I think and usb to your laptop or computer, to record with.
Wouldn't the OpenScale setup work the same way and avoid the additional step? As far as I can tell, it can take the load cell data, amplify it, and convert it so the computer can record it in one product.

#### r66astro

##### Well-Known Member
TRF Supporter
Wouldn't the OpenScale setup work the same way and avoid the additional step? As far as I can tell, it can take the load cell data, amplify it, and convert it so the computer can record it in one product.
It can, a lot more work. Dataq is cheap and no programming. The other way you have to depend on libraries (or write your own) and writing your own sketch.

#### ttabbal

##### Well-Known Member
OpenScale + OpenLog is a complete setup. Dataq has the ADC, but you need to amplify the load cell voltages. Amps have historically been expensive. And you need a computer to read the data, I think. You would need to use one to read the data from OpenLog and plot it etc., but not in the field for logging the burn. Downside to that is you can't know at the time that you got the data you were after. So there is a benefit to having a display on site. OpenScale + a computer is basically the same, but you would need some software to interpret the data. From the description, the OpenScale just spits out a weight periodically. For graphs etc you would have to save the data, then load it into a plotting program, spreadsheet, whatever.

Another option is the Analog Devices eval board I mentioned above. It's $60, the ADC has an internal load cell amp, it's much faster, and the software they supply works well enough to read the data, save it out, and plot it on the screen. There is a downside, it only works with older Windows systems. You need Win7 or below. I use an old netbook on XP. It also doesn't work in a VM, at least not on Virtualbox anyway. #### JackO ##### MIT AeroAstro Undergrad -- HSRPL Founder It can, a lot more work. Dataq is cheap and no programming. The other way you have to depend on libraries (or write your own) and writing your own sketch. Aren't there programs available online to do this? If not, we will go with Dataq. We want the simplest setup possible to get us thrust measurements. #### sghioto ##### Well-Known Member I second the Dataq method as I have been using one for years. The latest starter kit from Dataq is only 59.00 and load cell amps from China under 20.00 on Ebay are common. So for less then 100.00 you should be all set. Again the advantage is you can record the thrust measurement, plot the graph in real time using any laptop and no programming required. Steve G #### G_T ##### Well-Known Member There is a learning curve making the first good motor, a second learning curve making consistent good motors, and another learning curve scaling up to larger motors. Does your group have at least one member with proven expertise making consistent solid propellant M or higher motors that match design specs? That is a starting point for moving up towards the P realm (in SMALLER steps); nothing more. Just being able to burn a few M motors without them going CATO doesn't cut it. There is a lot more to learn to make a P. A P has 8 times the energy in twice the size. The larger the motor, the less forgiving of inconsistency, ignorance, and error. If the starting point isn't already met, the learning curve is measured in years IMHO. I've been through the learning curve to where I'd be comfortable designing and fabricating a P motor, though O is the largest I have done. Ms are my little test motors. To get to where I felt like I had a clue (doesn't mean there aren't some things I should know but don't) has taken some years, a few hundred pounds of AP, lots of static tests and flight tests, hundreds of research papers, private emails and discussions, plus a mentor who already made large motors routinely. I expect any motor of mine to have propellant which is close to theoretical density, reproducibly so, to produce the thrust curve to design, and have a burn duration matching target to 0.1 seconds. I expect the resulting altitude when flown to hit within a couple percent of design, and usually it is better than that. You CAN build a large motor by overbuilding it and designing conservative. But you still have to get far enough along to know what that means, in specifics. I'm not trying to dissuade you, but among other things, solids are not liquids. Does your group have someone experienced in solids? The health and safety issues are not quite the same, and there are certainly manufacturing, handling, storage, and transport differences. Gerald #### dmo ##### Active Member The OpenScale module is probably just fine for what it was designed for, but I would not consider using it for a motor test stand. The HX711 it uses is only capable of 80 sps which is way to slow. #### NateLowrie ##### Well-Known Member There is a learning curve making the first good motor, a second learning curve making consistent good motors, and another learning curve scaling up to larger motors. Does your group have at least one member with proven expertise making consistent solid propellant M or higher motors that match design specs? That is a starting point for moving up towards the P realm (in SMALLER steps); nothing more. Just being able to burn a few M motors without them going CATO doesn't cut it. There is a lot more to learn to make a P. A P has 8 times the energy in twice the size. The larger the motor, the less forgiving of inconsistency, ignorance, and error. If the starting point isn't already met, the learning curve is measured in years IMHO. I've been through the learning curve to where I'd be comfortable designing and fabricating a P motor, though O is the largest I have done. Ms are my little test motors. To get to where I felt like I had a clue (doesn't mean there aren't some things I should know but don't) has taken some years, a few hundred pounds of AP, lots of static tests and flight tests, hundreds of research papers, private emails and discussions, plus a mentor who already made large motors routinely. I expect any motor of mine to have propellant which is close to theoretical density, reproducibly so, to produce the thrust curve to design, and have a burn duration matching target to 0.1 seconds. I expect the resulting altitude when flown to hit within a couple percent of design, and usually it is better than that. You CAN build a large motor by overbuilding it and designing conservative. But you still have to get far enough along to know what that means, in specifics. I'm not trying to dissuade you, but among other things, solids are not liquids. Does your group have someone experienced in solids? The health and safety issues are not quite the same, and there are certainly manufacturing, handling, storage, and transport differences. Gerald Second what Gerald said. Make sure you have your safety procedures and processing procedures drawn up and reviewed by qualified people. I think it's awesome you are pursuing this, but we all want you to be safe while doing so. If you want a double check, send us the procedures and we'll look them over. Just make sure you have all your ducks in a row before you start about what steps you are going to perform and how exactly you are going to do them, because doing something stupid like mixing powders dry is highly hazardous at best and fatal at worst. #### ttabbal ##### Well-Known Member The OpenScale module is probably just fine for what it was designed for, but I would not consider using it for a motor test stand. The HX711 it uses is only capable of 80 sps which is way to slow. You know, I thought so too, but someone posted a curve generated by one and it's not bad. I think for a larger project and for pressure it's worth going faster though. #### JackO ##### MIT AeroAstro Undergrad -- HSRPL Founder There is a learning curve making the first good motor, a second learning curve making consistent good motors, and another learning curve scaling up to larger motors. Does your group have at least one member with proven expertise making consistent solid propellant M or higher motors that match design specs? That is a starting point for moving up towards the P realm (in SMALLER steps); nothing more. Just being able to burn a few M motors without them going CATO doesn't cut it. There is a lot more to learn to make a P. A P has 8 times the energy in twice the size. The larger the motor, the less forgiving of inconsistency, ignorance, and error. If the starting point isn't already met, the learning curve is measured in years IMHO. I've been through the learning curve to where I'd be comfortable designing and fabricating a P motor, though O is the largest I have done. Ms are my little test motors. To get to where I felt like I had a clue (doesn't mean there aren't some things I should know but don't) has taken some years, a few hundred pounds of AP, lots of static tests and flight tests, hundreds of research papers, private emails and discussions, plus a mentor who already made large motors routinely. I expect any motor of mine to have propellant which is close to theoretical density, reproducibly so, to produce the thrust curve to design, and have a burn duration matching target to 0.1 seconds. I expect the resulting altitude when flown to hit within a couple percent of design, and usually it is better than that. You CAN build a large motor by overbuilding it and designing conservative. But you still have to get far enough along to know what that means, in specifics. I'm not trying to dissuade you, but among other things, solids are not liquids. Does your group have someone experienced in solids? The health and safety issues are not quite the same, and there are certainly manufacturing, handling, storage, and transport differences. Gerald Gerald, Our primary mentor originally from Rocketdyne has half a decade of experience in all types of propulsion; he is a contractor doing propulsion work for Northrop Grumman, Rocketdyne, Boeing, Virgin Galactic, and SpaceX in its early days. He had the amateur rocketry record back in the 90's with his boosted dart launched to 48 miles. He taught a class in propellant and many of the guys building the SRB's enrolled (routinely got 98%theoretical density in this class without a vacuum). I've consulted with Derek Deville rather extensively on the motor design. We have mentors at Aerojet Rocketdyne, and we are designing conservatively to the extreme. Our test engine has a 1/4" steel casing; the radial bolts for the bulkhead will give out long before the steel. We'll conduct a full static test of the P engine months before launch. All personnel will be in the bunker at the Mojave Test Area during tests and launches. I of course have not built a motor this big, but I think I have taken the necessary steps to safely and confidently do so. Regards, Jack Oswald #### JackO ##### MIT AeroAstro Undergrad -- HSRPL Founder OpenScale + OpenLog is a complete setup. Dataq has the ADC, but you need to amplify the load cell voltages. Amps have historically been expensive. And you need a computer to read the data, I think. You would need to use one to read the data from OpenLog and plot it etc., but not in the field for logging the burn. Downside to that is you can't know at the time that you got the data you were after. So there is a benefit to having a display on site. OpenScale + a computer is basically the same, but you would need some software to interpret the data. From the description, the OpenScale just spits out a weight periodically. For graphs etc you would have to save the data, then load it into a plotting program, spreadsheet, whatever. Another option is the Analog Devices eval board I mentioned above. It's$60, the ADC has an internal load cell amp, it's much faster, and the software they supply works well enough to read the data, save it out, and plot it on the screen. There is a downside, it only works with older Windows systems. You need Win7 or below. I use an old netbook on XP. It also doesn't work in a VM, at least not on Virtualbox anyway.
Thanks for the comment. So Dataq +amplifier will allow me to record and log the data without anything else? Or would I still need to get the software for data-logging? Because if that is the case then OpenScale would accomplish the same for cheaper. I am absolutely willing to spend extra but only to lose a step in the process. Thanks again for the advice.

-Jack

#### JackO

##### MIT AeroAstro Undergrad -- HSRPL Founder
The OpenScale module is probably just fine for what it was designed for, but I would not consider using it for a motor test stand. The HX711 it uses is only capable of 80 sps which is way to slow.
Ok, that is definitely good to know; maybe Dataq then? This subject is definitely foreign to me.

#### JackO

##### MIT AeroAstro Undergrad -- HSRPL Founder
Second what Gerald said. Make sure you have your safety procedures and processing procedures drawn up and reviewed by qualified people. I think it's awesome you are pursuing this, but we all want you to be safe while doing so. If you want a double check, send us the procedures and we'll look them over. Just make sure you have all your ducks in a row before you start about what steps you are going to perform and how exactly you are going to do them, because doing something stupid like mixing powders dry is highly hazardous at best and fatal at worst.
Like I told Gerald, we have very qualified mentors. I'd be happy to send you guys the procedures; not entirely sure how to outside of the Research Forum. We cast the propellant last Saturday; here is the forum thread where I am slowly documenting our progress: https://www.rocketryforum.com/showthread.php?138502-Chaminade-Rocket-Club-High-Power-Rocket

#### JackO

##### MIT AeroAstro Undergrad -- HSRPL Founder
You know, I thought so too, but someone posted a curve generated by one and it's not bad. I think for a larger project and for pressure it's worth going faster though.
So what's the consensus? Is the OpenScale sufficient? Or should we use Dataq? I'd like to order the products later today if possible.

#### ttabbal

##### Well-Known Member
I wouldn't say there is a consensus.

Dataq + amp is the same as the openscale as far as collecting data. Both look to need a computer. The biggest differences are that the dataq is faster, and comes with software to capture and display the data. The downside is that you need an amplifier as well. The cheaper dataq is lower resolution as well, but that might not matter with the amp.

I've seen more people here use the dataq than HX711 based setups, so perhaps that could be called consensus...

#### JackO

##### MIT AeroAstro Undergrad -- HSRPL Founder
I wouldn't say there is a consensus.

Dataq + amp is the same as the openscale as far as collecting data. Both look to need a computer. The biggest differences are that the dataq is faster, and comes with software to capture and display the data. The downside is that you need an amplifier as well. The cheaper dataq is lower resolution as well, but that might not matter with the amp.

I've seen more people here use the dataq than HX711 based setups, so perhaps that could be called consensus...

Edward


text:
# Greatest historian ever

In genius studies, greatest historian ever refers to individuals with a sharpened ability, technique, or approach to recount historical events of humans, in a brilliant way.

Overview
The following is a work in progress listing of famous historians as found in the top 1000 genius rankings:
-
 IQ Historian IQ citations Description 1. —20 Henry Adams (1838-1918)Collected works: 12+ [RGM:811|1,310+] (SN:2) (CR:241)Spent over five decades attempting to explain history according to physical principals, physico-chemical principles generally; in his 9-volume History of United States America (1891), covering the years during the Jefferson administration (1801-1817), was penned with the sole intention to demonstrate causality in the course of human history, or as Adams would latter summarize “to satisfy himself whether, by severest process of stating, with the least possible comment, such facts as seemed sure, in such order as seemed rigorously consequent, he could fix for a familiar moment a necessary sequence of human movement”; his The Tendency of History (1896), attempted to describe history according to the second law of thermodynamics, albeit of the form of William Thomson's 1852 paper "On a Universal Tendency to the Dissipation of Mechanical Energy"; his The Rule of Phase Applied to History (1909) was reviewed by physicist Henry Bumstead, one of the few and only students of Willard Gibbs, the inventor of the phase rule; his A Letter to American Teachers of History (1910) gave sound advice for future generations of historians to start looking back at history scientifically, the way scientists look at reaction patterns between molecules, to view people as human molecules, and historical progress as governed by the laws of thermodynamics, just as is the historical progress of the universe. 2. —170 Edward Gibbon(1737-1794) $IQ_C \,$=180 (Cattell 1000:60) [RGM:495|1,400+] (Murray 4000:N/A) [HD:12] (FA:76) (GHE:2) (CR:18) English historian; noted for his six-volume The History of the Decline and Fall of the Roman Empire, peer reviewed by David Hume (1776), wherein he openly criticized organized religion, telling the story of the rise and fall of Roman, without the intervention of god, as was previously the standing historical modus operandi, but instead from the point of view of Rome’s decline as due the “disease of Christianity” spreading through the Roman empire and rotting it. 3. —213 Henry Buckle (1821-1862) (Library:21,000) (CR:34) English social physics themed historian;“The actions of men are in reality never inconsistent, but however capricious they may appear only form part of one vast system of universal order.”— Henry Buckle (1861), History of Civilization in England In his 3-volume History of Civilization in England, which was immediately praised by Maxwell and Boltzmann, he focused on solving the question of whether the actions of people operate by (a) fixed laws, (b) chance, or (c) supernatural interference, via physical science means, i.e. strong positivism. 4. —249 Herodotus (484-425BC) (Cattell 1000:202) [RGM:171|1,500+] (CR:51) was a Greek historian, aka “father of history” (Cicero, 50BC), noted for his digression on a number of thinkers, e.g. Thales, and topics, e.g. his use of the term “phoenix” for Ra, of Heliopolis creation myth, connection of Osiris with Dionysus (see: Osiris, Dionysus, and Bacchus), mummification, the Egyptian pig superstition behind Mark 5:1-17, among others 5. —349 Oswald Spengler(1880-1936) =170 (RGM:245|1,500+) (CR:15) was a German philosopher of history;“Today we live so cowed under the bombardment of this intellectual artillery that hardly anyone can attain to the inward detachment that is required for a clear view of the monstrous drama. The will-to-power operating under a pure democratic disguise has finished off its masterpiece so well that the object's sense of freedom is actually flattered by the most thorough-going enslavement that has ever existed.”— Ostwald Spengler (1923), The Decline of the West, Volume Two (pg. 461)His two-volume The Decline of the West (1922), wherein he outlines a Goethean-Nietzschean view of rise and falls of civilization, intermixed, supposedly, with entropy discussion. 6. —527 Bede(672-735) (Cattell 1000:878) [RGM:615|1,500+] (CR:15) English historian and monk; noted for being one of the first to connect John the Baptist with the Aquarius constellation; recounted the “warm sparrow” anecdote Christianity adoption in England; his Ecclesiastical History of the English People (1731) gained him the epitaph as the “father of English history”. 7. —569 Thucydides (460-395BC) (Cattell 1000:72) [RGM:175|1,500+] (FA:14) (CR:3) Greek historian and general; “Modern criticism rightly regards Thucydides as the model of the rational and objective historian who made it his first duty to leave the finger of god out of history and to tidy up the mundane events of the human drama. The contrast between his work with respect to the gods and the histories of his predecessor Herodotus and his successor Xenophon is plain; it is a contrast so great, in fact, as to provoke the observation from some (e.g. K. J. Dover) that Thucydides may well have been an atheist.”— Borimir Jordan (1986), “Religion in Thucydides”noted for his participation in and later secular recounting of the Peloponnesian War (431-404BC), wherein the Athenians battled the Spartans; considered, by some (Ѻ), to be the father of “scientific history”, i.e. history based on evidence-gathering and analysis of cause and effect. 8. —611 Livy(59BC-17AD) (Cattell 1000:119) [RGM:N/A|1,310+] (Murray 4000:N/A) Roman historian; noted for his History of Rome (9BC), wherein he discusses the Mucius Scaevola hand burning legend (famously tested by Jean Rousseau and Friedrich Nietzsche in youth); Thomas Jefferson, in his recommended education letter to his nephew Peter Carr, says that one should read the Bible as one “would read Livy or Tacitus”; read by John Mill (Ѻ) between ate 8 and 12; cited in the silent historians problem 9. Tacitus(58-120AD) (Cattell 1000:294) (CR:11) Roman senator, public official, historian of the Roman Empire, who is noted as someone oft-cited having referred to the punishment of "Christians" by Nero and the execution of their founder "Christus" by Pontius Pilate; considered by Thomas Jefferson to be a good historian to read 10. —614 Josephus (c.37-100AD) (Cattell 1000: 645) (CR:7) Roman-Jewish historian, scholar, and hagiographer, oft-cited in the silent historians problem on whether or not he refers to a "Jesus" as the "Christ", a detail that was “unknown” to Origen (c.230AD), who had copies of his works; he also conjectured that the Jewish exodus was based on the 1300BC expulsion of the Hyksos from Egypt 11. Manetho (c.300-250BC) Egyptian historian and priest, centered in Heliopolis, noted for his History of Egypt, written, in Greek, for Ptolemy II (reign: 283-246BC), based on materials collected in the library of the priesthood of Ra; his division of the rulers of ancient Egypt into 30 dynasties is still used as the basic framework for ancient Egyptian history. (Ѻ)