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In the middle of last century, Bondi and his coworkers proposed an out goingboundary condition for the Einstein equations. Based on such boundary conditionthe authors theoretically solved the puzzle of the existence problem ofgravitational wave. Currently many works on gravitational wave source modelinguse Bondi's result to do the analysis including the gravitational wavecalculation in numerical relativity. Recently more and more observations implythat the Einstein equations should be modified with an nonzero cosmologicalconstant. In this work we use Bondi's original method to treat the Einsteinequations with cosmological constant for theoretical curiosity. We find thatthe gravitational wave does not essentially exist if Bondi's original out goingboundary condition is imposed. We find this unphysical result is due to thenon-consistency between Bondi's original out going boundary condition and theEinstein equations with cosmological constant. Based on theoretical analysis,we propose an new Bondi-type out going boundary condition for the Einsteinequations with cosmological constant. With this new boundary condition, thegravitational wave behavior for the Einstein equations with cosmologicalconstant is similar to the Einstein equations without cosmological constant. Inparticular, when the cosmological constant goes to zero, the behavior of theEinstein equations without cosmological constant is recovered. Together withgravitational wave, electromagnetic wave is also analyzed. The boundaryconditions for electromagnetic wave does not depend on the cosmologicalconstant.

Note on the gravitational and electromagnetic radiation for the Einstein-Maxwell equations with cosmological constant