Ultrafast electromagnetic waves radiated from semiconductor material under high electric fields and photoexcited by femtosecond laser pulses have been recorded by using terahertz time domain spectroscopy (THz-TDS). The waveforms of these electromagnetic waves reflect the dynamics of the photoexcited carriers in the semiconductor material, thus, THz-TDS provides a unique opportunity to observe directly the temporal and spatial evolutions of non-equilibrium transport of carriers within sub-picosecond time scale. We report on the observed THz emission waveforms emitted from GaAs by using a novel technology, the time domain THz electro-optic (EO) sampling, which has a bipolar feature, i.e., an initial positive peak and a subsequent negative dip that arises from its velocity overshoot. The initial positive peak has been interpreted as electron acceleration in the bottom of valley in GaAs, where electrons have a light effective mass. The subsequent negative dip has been attributed to intervalley transfer from to X and L valleys. Furthermore, the power dissipation spectra of the bulk GaAs in THz range are also investigated by using the Fourier transformation of the time domain THz traces. From the power dissipation spectra, the cutoff frequency for negative power dissipation (i.e., gain) under step electric field in the bulk GaAs can also be obtained. The cutoff frequency for the gain gradually increases with increasing electric fields up to 50 kV/cm and achieves saturation at approximately 1 THz at 300 K. Furthermore, based on the temperature dependence of the cutoff frequency, we find that this cutoff frequency is governed by the energy relaxation process of electrons from L to valley via successive optical phonon emission.