kb/data/en.wikipedia.org/wiki/Curved_spacetime-3.md

title	chunk	source	category	tags	date_saved	instance
Curved spacetime	4/5	https://en.wikipedia.org/wiki/Curved_spacetime	reference	science, encyclopedia	2026-05-05T11:15:02.570747+00:00	kb-cron

      T
      
        i
        j
      
    
  

{\displaystyle T^{ij}}

are the rates of flow of the i-component of momentum per unit area in the j-direction. Even if there is no bulk motion, random thermal motions of the particles will give rise to momentum flow, so the i = j terms (green) represent isotropic pressure, and the i ≠ j terms (blue) represent shear stresses. One important conclusion to be derived from the equations is that, colloquially speaking, gravity itself creates gravity. Energy has mass. Even in Newtonian gravity, the gravitational field is associated with an energy, ⁠

    E
    =
    m
    g
    h
    ,
  

{\displaystyle E=mgh,}

⁠ called the gravitational potential energy. In general relativity, the energy of the gravitational field feeds back into creation of the gravitational field. This makes the equations nonlinear and hard to solve in anything other than weak field cases. Numerical relativity is a branch of general relativity using numerical methods to solve and analyze problems, often employing supercomputers to study black holes, gravitational waves, neutron stars and other phenomena in the strong field regime.

=== Energy-momentum ===

In special relativity, mass-energy is closely connected to momentum. Just as space and time are different aspects of a more comprehensive entity called spacetime, mass–energy and momentum are merely different aspects of a unified, four-dimensional quantity called four-momentum. In consequence, if mass–energy is a source of gravity, momentum must also be a source. The inclusion of momentum as a source of gravity leads to the prediction that moving or rotating masses can generate fields analogous to the magnetic fields generated by moving charges, a phenomenon known as gravitomagnetism.

It is well known that the force of magnetism can be deduced by applying the rules of special relativity to moving charges. (An eloquent demonstration of this was presented by Feynman in volume II, chapter 13–6 of his Lectures on Physics, available online.) Analogous logic can be used to demonstrate the origin of gravitomagnetism. In Fig. 5-7a, two parallel, infinitely long streams of massive particles have equal and opposite velocities −v and +v relative to a test particle at rest and centered between the two. Because of the symmetry of the setup, the net force on the central particle is zero. Assume ⁠

    v
    ≪
    c
  

{\displaystyle v\ll c}

⁠ so that velocities are simply additive. Fig. 5-7b shows exactly the same setup, but in the frame of the upper stream. The test particle has a velocity of +v, and the bottom stream has a velocity of +2v. Since the physical situation has not changed, only the frame in which things are observed, the test particle should not be attracted towards either stream. It is not at all clear that the forces exerted on the test particle are equal. (1) Since the bottom stream is moving faster than the top, each particle in the bottom stream has a larger mass energy than a particle in the top. (2) Because of Lorentz contraction, there are more particles per unit length in the bottom stream than in the top stream. (3) Another contribution to the active gravitational mass of the bottom stream comes from an additional pressure term which, at this point, we do not have sufficient background to discuss. All of these effects together would seemingly demand that the test particle be drawn towards the bottom stream. The test particle is not drawn to the bottom stream because of a velocity-dependent force that serves to repel a particle that is moving in the same direction as the bottom stream. This velocity-dependent gravitational effect is gravitomagnetism. Matter in motion through a gravitomagnetic field is hence subject to so-called frame-dragging effects analogous to electromagnetic induction. It has been proposed that such gravitomagnetic forces underlie the generation of the relativistic jets (Fig. 5-8) ejected by some rotating supermassive black holes.

=== Pressure and stress === Quantities that are directly related to energy and momentum should be sources of gravity as well, namely internal pressure and stress. Taken together, mass-energy, momentum, pressure and stress all serve as sources of gravity: Collectively, they are what tells spacetime how to curve. General relativity predicts that pressure acts as a gravitational source with exactly the same strength as mass–energy density. The inclusion of pressure as a source of gravity leads to dramatic differences between the predictions of general relativity versus those of Newtonian gravitation. For example, the pressure term sets a maximum limit to the mass of a neutron star. The more massive a neutron star, the more pressure is required to support its weight against gravity. The increased pressure, however, adds to the gravity acting on the star's mass. Above a certain mass determined by the Tolman–Oppenheimer–Volkoff limit, the process becomes runaway and the neutron star collapses to a black hole. The stress terms become highly significant when performing calculations such as hydrodynamic simulations of core-collapse supernovae. These predictions for the roles of pressure, momentum and stress as sources of spacetime curvature are elegant and play an important role in theory. In regards to pressure, the early universe was radiation dominated, and it is highly unlikely that any of the relevant cosmological data (e.g. nucleosynthesis abundances, etc.) could be reproduced if pressure did not contribute to gravity, or if it did not have the same strength as a source of gravity as mass–energy. Likewise, the mathematical consistency of the Einstein field equations would be broken if the stress terms did not contribute as a source of gravity.

== Experimental test of the sources of spacetime curvature ==

=== Definitions: Active, passive, and inertial mass === Bondi distinguishes between different possible types of mass: (1) active mass (

      m
      
        a
      
    
  

{\displaystyle m_{a}}

) is the mass which acts as the source of a gravitational field; (2)passive mass (

      m
      
        p
      
    
  

{\displaystyle m_{p}}

) is the mass which reacts to a gravitational field; (3) inertial mass (

      m
      
        i
      
    
  

{\displaystyle m_{i}}

) is the mass which reacts to acceleration.