For men, B.E.E. = + ( x kg) + ( x cm) - ( x age) For women, B.E.E. = + ( x kg) + ( x cm) - ( x age) Total Caloric Requirements equal the B.E.E. multiplied by the sum of the stress and activity factors. The fraction on the left-hand side of this equation is the rocket's mass ratio by definition. This equation indicates that a ?v of times the exhaust velocity requires a mass ratio of. For instance, for a vehicle to achieve a of times its exhaust velocity would require a mass ratio of (approximately ).
In aerospace engineeringmass ratio is a measure of the efficiency of a rocket. It describes how what bottles are best for breast milk more massive the vehicle is with propellant than without; that is, the ratio of the rocket's wet mass vehicle plus contents plus propellant to its dry mass vehicle plus contents.
A more efficient rocket design requires less propellant to achieve a given goal, and would therefore have a lower mass ratio; however, for any given efficiency a higher mass ratio typically permits the vehicle to achieve higher delta-v. Typical multistage rockets have mass ratios in the range from 8 to The Space Shuttlefor example, has a mass ratio around The definition arises naturally from Tsiolkovsky's rocket equation :. Sutton defines the mass ratio inversely as: . Zubrin, Robert Entering Space: Creating a Spacefaring Civilization.
ISBN From Wikipedia, the free encyclopedia. For mass ratio in chemistry, see stoichiometry. Orbital mechanics. Orbital elements. Types of two-body orbitsby eccentricity. Circular orbit Elliptic orbit Transfer orbit Hohmann transfer orbit Bi-elliptic transfer orbit.
Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation. Celestial mechanics. Gravitational influences. Barycenter Hill sphere Perturbations Sphere of influence. N-body orbits. Lagrangian points Halo orbits. Engineering and efficiency. Preflight engineering. Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation.
Gravity assist Oberth effect. Sutton, Oscar Biblarz. Categories : How do electors vote in federal elections Mass Ratios. Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file. Download as PDF Printable version.
Types of two-body orbitsby eccentricity Circular orbit Elliptic orbit Transfer orbit Hohmann transfer orbit Bi-elliptic transfer orbit Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit.
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation. Gravitational influences Barycenter Hill sphere Perturbations Sphere of influence. N-body orbits Lagrangian points Halo orbits Lissajous orbits Lyapunov orbits.
Preflight engineering Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation. Efficiency measures Gravity assist Oberth effect.
The EFE is a tensor equation relating a set of symmetric 4 ? 4 tensors. Each tensor has 10 independent components. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge-fixing degrees of freedom, which correspond to the freedom to. Jan 18, · Body mass index is a common tool for deciding whether a person has an appropriate body datmixloves.com measures a person’s weight in relation to .
In the general theory of relativity the Einstein field equations EFE ; also known as Einstein's equations relate the geometry of spacetime to the distribution of matter within it. The equations were first published by Einstein in in the form of a tensor equation  which related the local spacetime curvature expressed by the Einstein tensor with the local energy, momentum and stress within that spacetime expressed by the stress—energy tensor.
Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations , the EFE relate the spacetime geometry to the distribution of mass—energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress—energy—momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of non-linear partial differential equations when used in this way.
The solutions of the EFE are the components of the metric tensor. The inertial trajectories of particles and radiation geodesics in the resulting geometry are then calculated using the geodesic equation. As well as implying local energy—momentum conservation, the EFE reduce to Newton's law of gravitation in the limit of a weak gravitational field and velocities that are much less than the speed of light.
Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied since they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the spacetime as having only small deviations from flat spacetime , leading to the linearized EFE.
These equations are used to study phenomena such as gravitational waves. The Einstein field equations EFE may be written in the form:  . This is a symmetric second-degree tensor that depends on only the metric tensor and its first- and second derivatives. The Einstein gravitational constant is defined as  .
The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress—energy—momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress—energy—momentum determines the curvature of spacetime.
These equations, together with the geodesic equation ,  which dictates how freely falling matter moves through spacetime, form the core of the mathematical formulation of general relativity. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge-fixing degrees of freedom , which correspond to the freedom to choose a coordinate system. Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions.
The equations are more complex than they appear. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations. Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative:.
Taking the trace with respect to the metric of both sides of the EFE one gets. Solving for R and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form:. Reversing the trace again would restore the original EFE. Einstein then included the term with the cosmological constant to allow for a universe that is not expanding or contracting.
This effort was unsuccessful because:. The inclusion of this term does not create inconsistencies. For many years the cosmological constant was almost universally assumed to be zero. Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side and incorporated as part of the stress—energy tensor:. The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign.
This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity.
Contracting the differential Bianchi identity. The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition. The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields , and charge and current distributions i.
The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation. In fact, the constant G appearing in the EFE is determined by making these two approximations. The orbit of a free-falling particle satisfies. In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form. To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero.
Applying these simplifying assumptions to the spatial components of the geodesic equation gives. This will reduce to its Newtonian counterpart, provided.
So this simplifies to. The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution. Additionally, the covariant Maxwell equations are also applicable in free space:.
The first equation asserts that the 4- divergence of the 2-form F is zero, and the second that its exterior derivative is zero. This is often taken as equivalent to the covariant Maxwell equation from which it is derived. The solutions of the Einstein field equations are metrics of spacetime. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime.
As the field equations are non-linear, they cannot always be completely solved i. For example, there is no known complete solution for a spacetime with two massive bodies in it which is a theoretical model of a binary star system, for example.
However, approximations are usually made in these cases. These are commonly referred to as post-Newtonian approximations. Even so, there are several cases where the field equations have been solved completely, and those are called exact solutions.
The study of exact solutions of Einstein's field equations is one of the activities of cosmology. It leads to the prediction of black holes and to different models of evolution of the universe. One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum. As discussed by Hsu and Wainwright,  self-similar solutions to the Einstein field equations are fixed points of the resulting dynamical system.
New solutions have been discovered using these methods by LeBlanc  and Kohli and Haslam. The nonlinearity of the EFE makes finding exact solutions difficult.
One way of solving the field equations is to make an approximation, namely, that far from the source s of gravitating matter, the gravitational field is very weak and the spacetime approximates that of Minkowski space. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the Minkowski metric , ignoring higher-power terms.
This linearization procedure can be used to investigate the phenomena of gravitational radiation. Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be written. Substituting this definition of the inverse of the metric into the equations then multiplying both sides by a suitable power of det g to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives.
The action from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields. See General relativity resources. From Wikipedia, the free encyclopedia.
Field equations in general relativity. Introduction History. Fundamental concepts. Principle of relativity Theory of relativity General covariance Simultaneity Relativity of simultaneity Relative motion Event Frame of reference Inertial frame of reference Mass Inertial mass Rest frame Center-of-momentum frame Curvature Geodesic Geon Equivalence principle Mass in general relativity Mass—energy equivalence Invariant Invariant mass Spacetime symmetries Special relativity Doubly special relativity de Sitter invariant special relativity Scale relativity Speed of light Time derivative Proper time Proper length Length contraction Action at a distance Principle of locality Riemannian geometry Energy condition.
Equations Formalisms. Birkhoff's theorem Geroch's splitting theorem Goldberg—Sachs theorem Lovelock's theorem No-hair theorem Penrose—Hawking singularity theorems Positive energy theorem. Special relativity General relativity. Spacetime concepts. Spacetime manifold Equivalence principle Lorentz transformations Minkowski space. General relativity. Introduction to general relativity Mathematics of general relativity Einstein field equations. Classical gravity. Introduction to gravitation Newton's law of universal gravitation.
Relevant mathematics. Four-vector Derivations of relativity Spacetime diagrams Differential geometry Curved spacetime Mathematics of general relativity Spacetime topology.
Main article: Cosmological constant. See also: Maxwell's equations in curved spacetime. Main article: Solutions of the Einstein field equations. Main article: Linearized gravity. Einstein—Hilbert action Equivalence principle Exact solutions in general relativity General relativity resources History of general relativity Hamilton—Jacobi—Einstein equation Mathematics of general relativity Numerical relativity Ricci calculus.
Annalen der Physik. Bibcode : AnP Archived from the original PDF on Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin : —